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Department of Physics, Deshbandhu College (University of Delhi), New Delhi Class: B.Sc Department of Physics, Deshbandhu College (University of Delhi), New delhi Class: B.Sc. (H) Semester-II (GE) Name of Paper: Mechanics Name of Teacher: Dr. Vikram Verma Differential Equation An ODE contains only ordinary derivatives (no partial derivatives) and describes the relationship between these derivatives of the dependent variable, usually called y, with respect to the independent variable, usually called x. The solution to such an ODE is therefore a function of x and is written y(x). The order of an ODE is simply the order of the highest derivative it contains. Thus equations containing dy/dx, but no higher derivatives, are called first order, those containing d2y/dx2 are called second order and so on. The degree of an ODE is the power to which the highest-order derivative is raised, after the equation has been rationalised to contain only integer powers of derivatives. Hence the ODE A general linear ODE of order n has the form If f(x) = 0 then the equation is called homogeneous; otherwise it is inhomogeneous. First-degree first-order equations: are several different types of first-degree first-order ODEs that are of interest in the physical sciences. Page 1 of 15 Department of Physics, Deshbandhu College (University of Delhi), New delhi Class: B.Sc. (H) Semester-II (GE) Name of Paper: Mechanics Name of Teacher: Dr. Vikram Verma Solution of Differential Equations 1. Separation of Variable Method: A separable-variable equation is one which may be written in the conventional form where f(x) and g(y) are functions of x and y respectively, including cases in which f(x) or g(y) is simply a constant. Rearranging this equation so that the terms depending on x and on y appear on opposite sides (i.e. are separated), and integrating, we obtain . Solution method- Factorise the equation so that it becomes separable. Rearrange it so that the terms depending on x and those depending on y appear on opposite sides and then integrate directly. Remember the constant of integration, which can be evaluated if further information is given. Exercise: Page 2 of 15 Department of Physics, Deshbandhu College (University of Delhi), New delhi Class: B.Sc. (H) Semester-II (GE) Name of Paper: Mechanics Name of Teacher: Dr. Vikram Verma 2. Homogeneous Differential Equations: Homogeneous equation are ODEs that may be written in the form where A(x, y) and B(x, y) are homogeneous functions of the same degree. A function f(x, y) is homogeneous of degree n if, for any λ, it obeys For example, if A = x2y – xy2 and B = x3 + y3 then we see that A and B are both homogeneous functions of degree 3. In general, for functions of the form of A and B, we see that for both to be homogeneous, and of the same degree, we require the sum of the powers in x and y in each term of A and B to be the same. The RHS of a homogeneous ODE can be written as a function of y/x. The equation may then be solved by making the substitution y = vx, so that This is now a separable equation and can be integrated directly to give Solution method: Check to see whether the equation is homogeneous. If so, make the substitution y = vx, separate variables and then integrate directly. Finally replace v by y/x to obtain the solution. Page 3 of 15 Department of Physics, Deshbandhu College (University of Delhi), New delhi Class: B.Sc. (H) Semester-II (GE) Name of Paper: Mechanics Name of Teacher: Dr. Vikram Verma 3. Equations Reducible to Homogeneous Form: Let us consider a differential equation of the form (3.1) where a, b, c, e, f and g are all constants. The differential equation (3.1) can be reduced to homogeneous form by the substitution x = X + α and y = Y + β, (3.2) where α and β are constants and can be found from (3.3) (3.4) The eq. (3.1) reduces to (3.5) which is homogeneous and can be solved by the method discussed above. Case of failure: If a/e = b/f then (3.3) and (3.4) are not independent and so cannot be solved uniquely for α and β. In this case substitute a/e = b/f = 1/m ---> e = am & f = bm and then the given equation becomes dy ax by c . dx m(ax by) g Now put ax+by = z and apply the method of variable separable. Which is homogeneous ODE and can be solve by substituting y = vx, dy/dx = v + x(dv/dx). Page 4 of 15 Department of Physics, Deshbandhu College (University of Delhi), New delhi Class: B.Sc. (H) Semester-II (GE) Name of Paper: Mechanics Name of Teacher: Dr. Vikram Verma 4. Exact Differential Equations: An exact first-degree first-order ODE is one of the form (4.1) (4.2) From which we obtain, (4.3) (4.4) (4.5) If (4.5) holds then (4.1) can be written dU(x, y) = 0, which has the solution U(x, y) = c, where c is a constant and from (4.3) U(x, y) is given by (4.6) The function F(y) can be found from (4.4) by differentiating (4.6) with respect to y and equating to B(x, y). Page 5 of 15 Department of Physics, Deshbandhu College (University of Delhi), New delhi Class: B.Sc. (H) Semester-II (GE) Name of Paper: Mechanics Name of Teacher: Dr. Vikram Verma U F A(x, y)dx B(x, y) y y y F(y) [B(x, y) { A(x, y)dx}]dy (4.7) y Hence the solution the differential eq. (4.1) is given by A(x, y)dx [B(x, y) { A(x, y)dx}]dy C , (4.8) y where C is the constant of Integration. 5. Inexact Differential Equations: Equations that may be written in the form (5.1) are known as inexact equations. However, the differential A dx + B dy can always be made exact by multiplying an integrating factor μ(x, y), which obeys (5.2) For an integrating factor which is a function of both x and y, i.e. μ = μ(x, y), there exists no general method for finding it; in such cases it may sometimes be found by inspection. If, however, an integrating factor exists that is a function of either x or y alone then (5.2) can be solved to find it. Page 6 of 15 Department of Physics, Deshbandhu College (University of Delhi), New delhi Class: B.Sc. (H) Semester-II (GE) Name of Paper: Mechanics Name of Teacher: Dr. Vikram Verma 1 A B f (x)dx (1) If f (x) , then integrating factor will be (x) e . B y x 1 B A g(y)dy (2) If g(y) , then integrating factor will be (y) e . A x y 1 (3) If A y f (xy)&B x g(xy) , then integrating factor will be . Ax By (4) If Mdx+Ndy = 0 is homogeneous equation and Mx+Ny ≠ 0, then integrating factor will be 1/(Mx+Ny). Exercise: Page 7 of 15 Department of Physics, Deshbandhu College (University of Delhi), New delhi Class: B.Sc. (H) Semester-II (GE) Name of Paper: Mechanics Name of Teacher: Dr. Vikram Verma 6. Linear Differential Equations: A differential Equation of the form (6.1) is called Linear Differential Equation, where P and Q or either functions x only or constants. Such equations are special case of inexact ODEs and can be made exact by multiplying through Pdx by an appropriate integrating factor (I.F.) given by I.F.e . (6.2) In this case, however, the integrating factor is always a function of x alone and may be expressed Pdx in a particularly simple form (x)e . Solution method. Rearrange the given differential equation into the form (2.1) and multiply by the integrating factor μ(x) given by (2.2). The left- and right-hand sides can then be integrated directly. The required solution of linear diff. eq. (2.1) will be given by y I.F. (Q I.F.)dx C (6.3) Page 8 of 15 Department of Physics, Deshbandhu College (University of Delhi), New delhi Class: B.Sc. (H) Semester-II (GE) Name of Paper: Mechanics Name of Teacher: Dr. Vikram Verma 7. Bernoulli’s equation Bernoulli’s equation has the form (7.1) This equation is very similar in form to the linear equation (7.1), but is in fact non-linear due to 1-n the extra yn factor on the RHS. However, the equation can be made linear by substituting v = y and correspondingly (7.2) Substituting this into (7.1) and dividing both sides by yn, we find which is a linear equation equation and can be solved by using method discussed in section-6. Page 9 of 15 Department of Physics, Deshbandhu College (University of Delhi), New delhi Class: B.Sc. (H) Semester-II (GE) Name of Paper: Mechanics Name of Teacher: Dr. Vikram Verma Page 10 of 15 Department of Physics, Deshbandhu College (University of Delhi), New delhi Class: B.Sc. (H) Semester-II (GE) Name of Paper: Mechanics Name of Teacher: Dr. Vikram Verma Page 11 of 15 Department of Physics, Deshbandhu College (University of Delhi), New delhi Class: B.Sc. (H) Semester-II (GE) Name of Paper: Mechanics Name of Teacher: Dr. Vikram Verma Page 12 of 15 Department of Physics, Deshbandhu College (University of Delhi), New delhi Class: B.Sc.
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