Non homogeneous linear differential equations pdf

Continue Differential equations that are linear in relation to an unknown and its derivatives This article is about linear differential equations with one independent variable. For similar equations with two or more independent variables, see the linear in , a differential equation that is defined by linear in an unknown function and its derivatives, ie form equation a 0 (x) y 1 (x) y y 2 ( x) y - ⋯ a n ( h) y ( n) b (h) a_{0} (x) y'a_{1} (x)y'a_{2} (x)y''a_ s ,, where 0 (x) display a_{0} (x), ..., a n (x) displaystyle a_'n (x) and b (x) displaystyle b'x) are arbitrary different functions that should not be linear, and y, ... y (n) displaystyle y', ldots,y' (n) are consistent derivatives. This is the usual differential equation (ODE). A linear differential equation can also be a linear partial differential equation (PDE) if an unknown function depends on multiple variables, and the derivatives that appear in the equation are partial derivatives. A linear differential equation or a system of linear equations, so that related homogeneous equations have constant ratios, can be solved by quadratic, meaning that solutions can be expressed in terms of integrations. This is also true for a linear equation of order one, with non-permanent odds. The equation of about two or higher with a non-constant ratio cannot, in general, be solved by a square. For about two, Kovacic's algorithm allows you to decide whether there are solutions from the point of view of , and calculate them, if any. Solutions of linear differential equations with polynomial coefficients are called holonic functions. This class of functions is stable under amounts, products, differentiation, integration and contains many conventional functions and special functions such as , logarith, sinus, cosin, reverse trigonometry functions, error function, Bessel functions and hypergeometric functions. Their presentation by determining the differential equation and the initial conditions allows us to make the algorithmic (on these functions) the majority of calculus operations, such as calculating antiderivations, limits, asymptomatic expansion, and numerical evaluation with any accuracy, with a certified error associated. The basic terminology of the Higher Order of Derivatives, which appears in a different equation, is the order of the equation. The term b(x), which does not depend on an unknown function and its derivatives, is sometimes referred to as the constant term of the equation (similar to algebraic equations), even if the term is not a constant function. If the term is a permanent term function, the differential equation is considered homogeneous, as it is a homogeneous polynomial in an unknown function and its derivatives. The equation obtained by replacement, in the linear differential equation, the constant term zero function is a related homogeneous equation. The differential equation has constant odds if only permanent functions are displayed as coefficients in a related homogeneous equation. Solving the differential equation is a function that satisfies the equation. Solutions of a homogeneous linear differential equation form a vector space. In the usual case, this vector space has a finsuute dimension equal to the order of the equation. All solutions to the linear differential equation are by adding to a specific solution any solution to the associated homogeneous equation. Linear Home article: Differential operator The main differential order i statement is a display that displays any diffused function to its ith derivative, or, in the case of multiple variables, To one of its partial i order derivatives. This is usually denoted by D i d x i 'displaystyle (fracas)d'i'dx'i in the case of univariate functions, and ∂ i 1 - ⋯ i n ∂ x 1 i 1 ⋯ ∂ x n n'display Style frak (partial i_{1}) -cdot (partial i_ x_{1} i_{1}) i_{1}) cdots (partial x_ n'n'i_) in the case of n variable functions. The main differential operators include the derivative from order 0, which is a display of identification. A linear differential operator (short, in this article, as a linear operator or, to put it simply, an operator) is a linear combination of the main differential operators, with different functions as coefficients. In a disadvantagey case, the line operator has, therefore, Form 0 (x) 1 (x) d x y ⋯ n (x) d d d n n , displaystyle a_{0} (x) a_{1} (x) x x-Kdota (a_) (x) Frak (d'n'd'n'n'n'n'), where 0 (x), ... , n (x) displaystyle a_{0} (x), ldots,a_'n'(x) - these are different functions, and non-negative n is the order of the operator (if n (x) (displaystyle a_'n'(x) is not a zero function). L to f function is usually indicated by Lf or Lf(X) if you want to specify a variable (this should not be confused with multiplication). The linear differential operator is a linear operator because it displays the amounts and product scalar to the product of the same scalar. Since the sum of the two line operators is a linear operator, as well as the product (left) of the linear operator for different functions, linear differential operators form a vector space above real numbers or complex numbers (depending on the nature of the functions reviewed). They also form a free module over the of various functions. Teh operators allow you to write compactly for different equations: if L q a 0 ( x) 1 ( x) d x - ⋯ a n ( x) d n d x a_{0} n , a_{1} (x) Frak (d'dx'cdots) a_ is a linear differential operator, then the equation a 0 (x) y 1 (x) y y y 2 2 ( x) y - ⋯ n ( x) y (n) b ( x) a_{0}. x)y'a_{1} (x)y'a_{2} (x)y''cdots (a_'n'(x)y' (n) 'b'x) can be rewritten L y q b (x) . Ly'b(x). There may be several variations of this notation; in particular, variable differentiation may appear explicitly or not in y and right hand and equations such as L y b (x) b (x) displaystyle Ly(x)b(x) or L y b displaystyle Lyb The core of the linear differential operator is its core as a linear mapping, i.e. a vector-based solution space (uniform) , the karateodori theorem implies that under very mild conditions, the L nucleus is a vector space for measurement n, and that the solutions to the L y (x) b (x) equation ⋯ displaystyle Ly(x) 'b'b(x) have the shape of S 0 (x) , display style S_{0} (x)c_{1}S_{1}(x) cdots c_ n'S_ n'nn (x), where c 1, ..., c display c_{1}, ldots,c_ n are arbitrary numbers. Typically, the karateodori theorem hypotheses are met in the I interval if functions b, a 0, ... n'displaystyle b,a_{0}, ldots,a_ n' continuous in I, and there is a positive real number k such, that a n (x) displaystyle a_ (x) zgt;k for each x in I. A homogeneous equation with constant coefficients Homogeneous linear differential equation has constant odds if it has the form of 0 y and 1 ⋯y ) 0 display a_{0}y'a_{1}y'a_{2}y'dots (a_ n'y'(n) where 1, ... n 'displaystyle a_{1}, ldots, a_ n' are (real or complex) numbers. In other words, it has constant odds if it is determined by a line operator with constant odds. The study of these differential equations with constant coefficients goes back to , who introduced the exponential function e x displaystyle ex, which is a unique solution to the equation f f displaystyle f'f in such a way that f (0 ) . It follows that nth derivative e c x displaystyle ecx is c n e c x, displaystyle c'n'e'cx, and this makes it quite easy to solve homogeneous linear differential equations. Пусть 0 у й й 1 й й й й 2 у й ⋯ й y ( n ) 0 дисплей стиль a_{0}y a_{1}y'a_{2} '''cdots (a_'n'y'(n) - это однородное линейное дифференциальное уравнение с постоянными коэффициентами (то есть 0,..., n 'displaystyle ,a_ are real or complex numbers). Finding solutions to this equation that are shaped like e α x displaystyle ealpha x is equivalent to finding a constant α alpha display so that 0 e α x 1 α e α x 2 α 2 e α x ⋯ n α n n n α 0. {\displaystyle a_{0}e^{\alpha x}+a_{1}\alpha e^{\alpha x}+a_{2}\alpha ^{2}e^{\alpha x}+\cdots +a_{n}\alpha ^{n}e^{\alpha x}=0.} Factoring from e α x displaystyle ealpha x (which is never zero), shows that α display style alpha should be the root of the characteristic polynomial' 0 ⋯ 1 t a_{0} . a_{1}т a_{2}т {2}cdots (a_'n't'n'n' дифференциального уравнения, которое является левой стороной характерного уравнения 0 - 1 т - 2 т 2 - ⋯ - n t n 0. a_ {2} a_{2} a_{0} a_{1} When all these roots are different, the person has different solutions that are not necessarily real, even if the equation ratios are real. These solutions can be shown as linearly independent, given the determinant of Vandermonde values of these decisions on x No 0, ..., n - 1. Together they form the basis of the vector space of differential equation solutions (i.e. the nucleus of the differential operator). Example y 'y' 2 y' 2 y' - 2 y 'y'y'y'y'y'y'y'y'y'y'y'y'y'y'y'y'0 has a characteristic equation z 4' 2 z 3' 2 z 2 z 2 z 1' 0. Display style z'{4}- 2z'{3}2z'{2}-2z'1'0. Thus, the basis of the solution is e i x, e q i x, e x, x x. Display e'ix;e'-ix,;e'x,;x'x. The real basis of the solution is thus cos ⁡ x, sin ⁡ x, e x, x e x. . Displaystyle cos x; sin x, e'x,»;x'x». In the case where the characteristic polynomial has only simple roots, the preceding provides a full basis of the vector space of solutions. In the case of a few roots, more linear, independent solutions are needed to have a framework. They have the form of x k e α x , displaystyle x'ke'alpha x where k is a non-integrator, α displaystyle alpha is the root of the characteristic polynomial plurality of m, and k zlt; m. To prove that these functions are solutions, you can order that if the α alpha display is the root of the characteristic polynomy of plurality m, the characteristic polynomial can be considered as P (t) (t q α) Thus, the application of a differential equation statement is equivalent to the use of the first m time operator d d x - α , display frac d'dx-alpha, and then the operator, which has P as a characteristic polynomial. By exponential shear theorem, (d d x - α) (x k e α x x x 1 e α x , display on the left (frac d'dx-alpha x'right) kx'k-1'e'alpha x, and thus one gets zero after k No. 1 application d d x and α . How, according to the fundamental algebra theorem, the sum of polynomy roots multiplication is equal to the degree of polynomial, the number of above-mentioned solutions is equal to the order of differential equation, and these solutions form the basis of the vector space of solutions. In general, when the coefficients of the equation are real, it is usually more convenient to have a basis for solutions consisting of real functions. Such a basis can be derived from the previous basis, noting that if the ib is the root of the characteristic polynomial, then - ib also root, the same plurality. Thus, the real basis is obtained with the Euler formula, and the replacement of x k e (a i b) x 'displaystyle x'k'e'(a'ib)x's and x k (a i b) x e x x x cos ⁡ (b x) (displaystyle x'k'e'ax'cos (bx ⁡) Displaystil x'k'e'ax'sin (bx). The case of the second-order Homogeneous linear differential equation of the second order can be written in - y b y y 0 , displaystyle y'ay'by'0, and its characteristic polynomial is r 2 y r . Display style {2} If a and b are real, there are three cases for solutions, depending on the discrimination D - 2 and 4 b. Display style D'a {2}-4b. In all three cases, the overall solution depends on two arbitrary constants c 1 (display c_{1}) and c2 (display c_{2}). If D is 0, the characteristic polynomial has two different real root α display alpha and β beta display. In this case, the common solution is c 1 e α x 2 e β x. display c_{1}ealpha x'c_{2}e'beta x. If D No. 0, the characteristic polynomial has a double root - a / 2 displaystyle -a/2, and a common solution (c 1 - c 2 x ) e - x / 2 . Display (c_{1} c_{2}x)e-ax/2. If D zlt; 0, the characteristic polynomial has two complex conjugated root α ± β I display style alpha p.1 beta i and a common solution c 1 e (α and β i) x x x 2 e (α - β i) x , displaystyle c_{1}e (Alpha Beta i)x'c_{2}'x that can be rewritten in real terms using the Euler formula as an electronic α x (c 1 cos ⁡ (β x) - c 2 sin ⁡ (β x). Display style e-alpha-x (c_{1} cos (beta x) c_{2} sin (beta x)). Finding a solution y (x) d_{1} display style y(x) satisfying y ( 0) y' (0) (d_{2}), one equates the values of the aforementioned overall solution by 0 and its derivatives there to d 1 displaystyle d_{1} and d 2, displaystyle d_{2}, respectively. This results in a linear system of two linear equations in two unknown c1 c_{1} and c 2. Display style c_{2}. The solution to this system provides a solution for the so-called Cauchy problem, which indicates values of 0 to solve the DEA and its derivative. Not a homogeneous equation with constant coefficients Not a homogeneous n order equation with constant odds can be written y ( n) (x) - 1 g (n q 1) ⋯ (x) , displaystyle y (n) (x) a_{1} (n-1) (x)cdots (a_ n-1'y'a_ y', a n'displaystyle a_{1}, ldots, a_'n' are real or complex numbers, f is this function x, and y is an unknown function (for the sake of simplicity ,' (x) will be omitted in the next). There are several ways to solve this equation. The best method depends on the nature of the f function, which makes the equation not homogeneous. If f is a linear combination of exponential and sine-sooid functions, an exponential response formula can be used. If, more generally, f is a linear combination of form functions x n e a x'displaystyle x'n'e'ax, x n cos ⁡ a x 'displaystyle x'n'cos, and x n sin ⁡ a x'displaystyle x'n'sin and constant (which should not be the same in every semester) then the method of uncertain odds can be used. Even more common, the method of destruction is applied when the f satisfies homogeneous linear differential equation, usually a holomic function. The most common method is the constant variation, which is presented here. A common solution to the related homogeneous equation y (n) - 1 g (n - 1) - ⋯ - n - 1 a_{1}y a_'n-1'y'y'y'a_'n'y'0' - is y u 1 y 1 - ⋯ u y n , displaystyle yu_{1}y_{1}'c y_ u_s, where (y 1, ... , y n) displaystyle (y_{1}, 'ldots, y_'n) is the basis of vector space solutions, and you 1 , ... u n' displaystyle u_{1}, ldots,u_ n' are arbitrary constants. The constant variation method takes its name from the following idea. Instead of treating u 1, ... u n 'displaystyle u_{1}, ldots,u_'n' as constants, they can be considered as unknown functions that need to be defined for making a decision not a homogeneous equation. С этой целью один добавляет ограничения 0 й u 1 y y y 2 y y 2 - ⋯ u n 'y n 'displaystyle 0'u'{1}y_{1}'u'{2}y_{2}'cdot ⋯ y_s u n q y n 'displaystyle 0'u'{1}y''{1}'u'u'{2}y'{2}'cdot ⋯s n No 2 ) - ⋯ - u n n ( n ) , displaystyle 0'u'{1}y_{1}(n-2 {2}y_{2}) cdots u'n'n'y_'n'n'(n-2) that imply (by product and induction rule) y (i) - u 1 y 1 (i ⋯) y n (i) «displaystyle y»(i)»u_{1}y_{1}(i) »cdots »u_'n»y_ (i) » для i q 1, ..., n - 1, и y ( n ) u 1 y 1 ( n ) - ⋯ u n y n n ( n ) u 1 q y 1 ( n q 1 ) u y ( n q 1 ) u 2 y 2 ( n ⋯ q 1 ) (displaystyle y'(n) u_{1}y_{1} (n) cdots (u_'n'y_'n'(n)'u'{1}y_{1} (n-1)'u'{2}y_{2} (n-1 y_) Замена в исходном уравнении у и его производных этими выражениями, и с использованием того факта, что у 1 , ... , y n 'displaystyle y_{1},'ldots, y_'n ' являются решениями оригинального однородного уравнения, один получает f q u 1 y 1 ( n ⋯ q 1 ) «Дисплей стиль f'u'{1}y_{1}» (n-1 y_) » Это уравнение и вышеперечисленные с 0 , как левая сторона образу县т систему n линейных уравнений в u 1 , ... , u n » » displaystyle u'{1}, ldots ,u'n ', чьи кофффициенты известны функции (f, yi, и их производные). Эта система может быть решена любым методом линейной алгебры. Вычисление антидеривативов дает u 1 , ... , u n , «displaystyle u_{1}, ldots,u_'n», а затем y u 1 y 1 и ⋯ u n n n . (дисплей y'u_{1}y_{1}'cdots (u_-н-y_). По мере того как antiderivatives определены до добавления постоянн, одно находит снова что об县ее разрешение non-однородного уравнения сумма произвольного решения и об县его решения associated однородного уравнения. Уравнение первого порядка с переменными кофффициентами Пример Решения уравнения y (x) - y ( x) / x 3 x . on Дисплейстиль y'(x)y(x)/x'3x. Связанное с 县тим однородное уравнение y (x) - y (x) / x q 0 (displaystyle y'(x)-y (x)/x'0' дает y / y / q 1 / x , displaystyle y'/y'-1/x, то есть y q / x . (дисплей y'c/x.) Разделение исходного уравнения одним из ттих решений дает х у дисплей стиль xy'y'3x» {2}.» То есть (x y) - 3 х 2 , дисплей (xy) (xy)'3x-{2}, х й х х 3 с, дисплей xyx'x{3}'c, и й ( х ) х 2 с / х . «Дисплейстиль y(x)» x{2}'c/x.» Для исходного состояния y ( 1 ) - α , displaystyle y(1) альфа, один получает конкретное решение y ( х ) х 2 и α 1 х . «Дисплейстиль y(x)» {2} »фрак »альфа-1» x». Об县ая форма линейного обычного дифференциального уравнения порядка 1, после деления кофффициента y (x) (displaystyle y'(x), является: y ( х ) х ( х ) й ( х ( х ) г ( х ...... «Дисплейстиль y'(x)f(x)y(x) g (x).» Если уравнение однородно, т.е.g(x) мофно переписать и интегрировать: y q y q f, фурнал ⁡ y q k q f , «displaystyle »frac »y'f», qquad «»k'F, где k является произвольной константой интеграции и F q ∫ f d x «displaystyle »textstyle F», d. Таким обра зом, об县им решением однородного уравнения является y s e F , «displaystyle y» ce'F», где c q e k«displaystyle c» e'k arbitrary constant. For a common heterogeneous equation, you can multiply it by a reciprocal e-display e-F (solution of a homogeneous equation. Displaystyle y'e-f-yfe-f-F. As well as f e - F d d x (e - F) , displaystyle -fe'-f-tfrac (d'dx' left (e'-f'right) , the product rule allows you to rewrite the equation as d d x (y e - F) g e. Displaystyle frac d'dx on the left (ye'-F-right) So the common solution is y q c e F and e F ∫ g e q f q d x, displaystyle y'ceF'e'f'int ge'-F'dx, where c is a constant of integration, and F q ∫ f--th. Linear Differential Equation System Main article: Matrix differential equation System Linear differential equations consists of several linear differential equations, which include several unknown functions. Generally one limits the study to systems such as that the number of unknown functions equals the number of levels. The arbitrary linear conventional differential equation and the system of such equations can be transformed into a first-order system of linear differential equations by adding variables to all but higher-order derivatives. That is, if y, y (k) 'displaystyle y','y', 'ldots,' y's (k) appear in the equations, you can replace them with new unknown functions y 1, ... y k 'displaystyle y_{1},' ldots, y_'y_ y_ y_{1} k, which should satisfy the equations y, to - 1. A first-order linear system that has n unknown functions and n differential equations can usually be solved for derivative unknown functions. If it's not, it's a differential algebraic system, and that's a different theory. So the systems which are discussed here, have a form of y 1 (x) - b 1 (x) - 1, 1 (x) y 1 - ⋯ - 1, n (x) y n ⋮ y n (x) y 1 - ⋯ - n, n (n) y n. displaystyle (beginning) y_{1} (x) b_{1} (x) a_ a_ y_{1},n'(x) y_'n'vdots (y_'n'n'(x) b_'n'n'(x) a_ n ,1(x)y_{1}'cdots (y_{1} a_,n'(x)y_,'n'n,'end'aligned, where b n 'displaystyle b_'n' and i, j 'displaystyle a_'i,j' are functions x. In the notation matrix, this system can be written (omitting (x)) in the display-style matbf (mathbf) Method of solution is similar to the method of linear differential equation of the first order, but with complications, arising from non-commutative. Display-style matbf (A'mathbf) . be a homogeneous equation associated with the above matrix equation. Its solutions form a vector space measuring n, and therefore U (x) displaystyle U(x function matrix, which is not a zero function. If n No. 1, or A is a constant matrix, or more generally, if A differs and commutes with its derivatives, you can opt for U exponential from antiderivative B - ∫ A d x ⁡ displaystyle textstyle B ⁡'int Adx A. Displaystyle frak (d'dx)exp (B) A'exp (B) or an approximation method, such as magnus expansion. Knowing the U Matrix, common solution not homogeneous equation y (x) - U (x) y 0 u ( x) ∫ U - 1 ( x) b ( x) d x , Mathbf y_{0} (x)-int U'-1 (x) mathbf (x), dx, where the column matrix y 0 display mathstyle y_{0} is an arbitrary constant. If the original terms are given as y (x 0) - y 0 , displaystyle mathbf y (x_{0}) mathbf y ({0}), a solution that satisfies these initial conditions y ( x) - U ( x) U 1 ( x 0 ) y 0 u (x) ∫ x 0 x U 1 (t) b ( t) t . Display-style mathbf (x)U'-1 (x_{0}) Mathbf (y_{0}) U(x) int (x_{0}'x'U'-1'-t)-mathbf (t) (t), dt. that means solutions can be expressed from the point of view of integrals. that was initiated by Emile Pickard and Ernest Vesiot and the latest developments of which are called the theory of differential Galois. The impossibility of a square solution can be compared to the Abel-Ruffini theoretic, which states that the algebraic equation of a degree of at least five cannot, in general, be decided by radicals. This analogy extends to evidence methods and motivates the denomination of Galois's differential theory. As in the algebraic case, the theory allows you to decide which equations can be solved by quadratic, and if possible to solve them. However, for both theories, the necessary calculations are extremely difficult, even with the most powerful computers. However, the case of order 2 with rational odds was completely solved by Kovacic's algorithm. Koshi-Euler equations of the Koshi-Euler equations are examples of equations of any order, with variable coefficients that can be solved explicitly. These are the form equations x n y (n) (h) , n - 1 x n 1 y (n q 1) (x) - ⋯ 0 g (x) a_-1'x'y(n-1) (x) cdots (a_{0}y (x) 0, where 0, ... n No.1 displaystyle a_{0}, ldots,a_ n-1 are constant odds. Main article: The holonomic function of the holomonic function, also called the D-end function, is a function that is the solution to a homogeneous linear differential equation with polynomial coefficients. Most of the functions commonly considered in mathematics are holomonic or holonomic factors. In fact, holomic functions include , algebraic functions, logaritis, exponential function, sinus, cosin, hyperbolic sinus, hyperbolic cosin, reverse trigonomeric and reverse hyperbolic functions, and many special functions such as Bessel functions and hypergeometric functions. Holomonic functions have several closure properties; in particular, holomonic amounts, products, derivatives and holomic functions integrals. In addition, these closure properties are effective, in the sense that there are algorithms to calculate the differential outcome equation of any of these operations, knowing the differential equations of input. The usefulness of the concept of holonic functions leads to the zeilberger theorem that follows. A holomonic is a sequence of numbers that can be caused by a recurrence of the relationship with polynomial coefficients. Taylor's series odds at the point of holomic function form a holomic sequence. Conversely, if the sequence of power series ratios is bare-footed, the series determines the holonomic function (even if the convergence radius is zero). There are effective algorithms for both conversions, i.e. to calculate relapse ratios from the differential equation, and vice versa. It follows that, if it is possible to present (in the computer) the holonomic functions of their defining differential equations and initial conditions, most calculus operations can be done automatically on these functions, such as derivatives, uncertain and certain , rapid calculations of the Taylor series (due to the recidivism to its coefficients), a high-precision score with a certified associated approximation error, the limits of localization, and the singularity of the , identity proof, etc. 4 See also Continuous-repayment of the fourier mortgage converts the equation of linear difference Link Change Gershenfeld 1999, p.9 Motivation: In analogy to complete the square, we write the equation as y f y g displaystyle y'-fy'g, and try to refine the left side so it becomes in particular, we strive to integrate the factor h (x ) displaystyle h'h (x) in such a way that multiplying on it makes the left side an equal derivative of the h ystyle hy i'm a h y' h f y (h y) . This means that h x x x x h'-hf, so h ∫. . as in the text. I'd like to see Seilberger, Doron. Holomonic systems are suitable for the identification of special functions. In the journal Computational and Applied Mathematics. 32.3 (1990): 321-368 - Benoit, A., Chizak, F., Darrasse, A., Herhold, S., Mezzarobba, M., Salvi, B. (2010, Sept. Dynamic Mathematical Function Dictionary (DDMF). At the International Congress on Mathematical Software (p. 35- 41). Springer, Berlin, Heidelberg. Birkoff, Garrett and Roth, Gian Carlo (1978), Ordinary Differential Equations, New York: John Wylie and Sons, Inc., ISBN 0-471-07411-X Gerschenfeld, Neil (1999), Nature Mathematical Modeling, Cambridge, UK: Cambridge University Press, ISBN 978-0-521-57095-4 Robinson, James C. (2004), Introduction to Conventional Differential Equations, Cambridge, UK: Cambridge University Press, ISBN 0-521-82650-0 External Links external Dynamic Dictionary function. Automatic and interactive study of many holomic features. Extracted from non homogeneous linear differential equations with constant coefficients. methods of solving non-homogeneous linear differential equations. examples of non homogeneous linear differential equations. nonhomogeneous second order linear differential equations. non homogeneous linear system of differential equations. solving non homogeneous first order linear differential equations. quantum algorithm for non-homogeneous linear partial differential equations. superposition principle for non homogeneous linear ordinary differential equations

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