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Variation of Parameters Overview An Example Double Check Further Discussion Variation of Parameters Bernd Schroder¨ logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Variation of Parameters y = yp + yh 2. Variation of Parameters is a way to obtain a particular solution of the inhomogeneous equation. 3. The particular solution can be obtained as follows. 3.1 Assume that the parameters in the solution of the homogeneous equation are functions. (Hence the name.) 3.2 Substitute the expression into the inhomogeneous equation and solve for the parameters. Overview An Example Double Check Further Discussion Variation of Parameters 1. The general solution of an inhomogeneous linear differential equation is the sum of a particular solution of the inhomogeneous equation and the general solution of the corresponding homogeneous equation. logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Variation of Parameters 2. Variation of Parameters is a way to obtain a particular solution of the inhomogeneous equation. 3. The particular solution can be obtained as follows. 3.1 Assume that the parameters in the solution of the homogeneous equation are functions. (Hence the name.) 3.2 Substitute the expression into the inhomogeneous equation and solve for the parameters. Overview An Example Double Check Further Discussion Variation of Parameters 1. The general solution of an inhomogeneous linear differential equation is the sum of a particular solution of the inhomogeneous equation and the general solution of the corresponding homogeneous equation. y = yp + yh logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Variation of Parameters 3. The particular solution can be obtained as follows. 3.1 Assume that the parameters in the solution of the homogeneous equation are functions. (Hence the name.) 3.2 Substitute the expression into the inhomogeneous equation and solve for the parameters. Overview An Example Double Check Further Discussion Variation of Parameters 1. The general solution of an inhomogeneous linear differential equation is the sum of a particular solution of the inhomogeneous equation and the general solution of the corresponding homogeneous equation. y = yp + yh 2. Variation of Parameters is a way to obtain a particular solution of the inhomogeneous equation. logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Variation of Parameters 3.1 Assume that the parameters in the solution of the homogeneous equation are functions. (Hence the name.) 3.2 Substitute the expression into the inhomogeneous equation and solve for the parameters. Overview An Example Double Check Further Discussion Variation of Parameters 1. The general solution of an inhomogeneous linear differential equation is the sum of a particular solution of the inhomogeneous equation and the general solution of the corresponding homogeneous equation. y = yp + yh 2. Variation of Parameters is a way to obtain a particular solution of the inhomogeneous equation. 3. The particular solution can be obtained as follows. logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Variation of Parameters (Hence the name.) 3.2 Substitute the expression into the inhomogeneous equation and solve for the parameters. Overview An Example Double Check Further Discussion Variation of Parameters 1. The general solution of an inhomogeneous linear differential equation is the sum of a particular solution of the inhomogeneous equation and the general solution of the corresponding homogeneous equation. y = yp + yh 2. Variation of Parameters is a way to obtain a particular solution of the inhomogeneous equation. 3. The particular solution can be obtained as follows. 3.1 Assume that the parameters in the solution of the homogeneous equation are functions. logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Variation of Parameters 3.2 Substitute the expression into the inhomogeneous equation and solve for the parameters. Overview An Example Double Check Further Discussion Variation of Parameters 1. The general solution of an inhomogeneous linear differential equation is the sum of a particular solution of the inhomogeneous equation and the general solution of the corresponding homogeneous equation. y = yp + yh 2. Variation of Parameters is a way to obtain a particular solution of the inhomogeneous equation. 3. The particular solution can be obtained as follows. 3.1 Assume that the parameters in the solution of the homogeneous equation are functions. (Hence the name.) logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Variation of Parameters Overview An Example Double Check Further Discussion Variation of Parameters 1. The general solution of an inhomogeneous linear differential equation is the sum of a particular solution of the inhomogeneous equation and the general solution of the corresponding homogeneous equation. y = yp + yh 2. Variation of Parameters is a way to obtain a particular solution of the inhomogeneous equation. 3. The particular solution can be obtained as follows. 3.1 Assume that the parameters in the solution of the homogeneous equation are functions. (Hence the name.) 3.2 Substitute the expression into the inhomogeneous equation and solve for the parameters. logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Variation of Parameters Z x 1 f (t) Z x 1 f (t) yp(x) = −y1(x) y2(t) dt + y2(x) y1(t) dt, x0 W(y1,y2)(t) a2(t) x0 W(y1,y2)(t) a2(t) where y1, y2 are linearly independent solutions of the 0 0 homogeneous equation and W(y1,y2) := y1y2 − y2y1. W is also called the Wronskian of y1 and y2. The Wronskian can also be represented as y1 y2 W(y1,y2) = det 0 0 . y1 y2 That’s it. Overview An Example Double Check Further Discussion Variation of Parameters For second order equations, we obtain the general formula logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Variation of Parameters where y1, y2 are linearly independent solutions of the 0 0 homogeneous equation and W(y1,y2) := y1y2 − y2y1. W is also called the Wronskian of y1 and y2. The Wronskian can also be represented as y1 y2 W(y1,y2) = det 0 0 . y1 y2 That’s it. Overview An Example Double Check Further Discussion Variation of Parameters For second order equations, we obtain the general formula Z x 1 f (t) Z x 1 f (t) yp(x) = −y1(x) y2(t) dt + y2(x) y1(t) dt, x0 W(y1,y2)(t) a2(t) x0 W(y1,y2)(t) a2(t) logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Variation of Parameters W is also called the Wronskian of y1 and y2. The Wronskian can also be represented as y1 y2 W(y1,y2) = det 0 0 . y1 y2 That’s it. Overview An Example Double Check Further Discussion Variation of Parameters For second order equations, we obtain the general formula Z x 1 f (t) Z x 1 f (t) yp(x) = −y1(x) y2(t) dt + y2(x) y1(t) dt, x0 W(y1,y2)(t) a2(t) x0 W(y1,y2)(t) a2(t) where y1, y2 are linearly independent solutions of the 0 0 homogeneous equation and W(y1,y2) := y1y2 − y2y1. logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Variation of Parameters The Wronskian can also be represented as y1 y2 W(y1,y2) = det 0 0 . y1 y2 That’s it. Overview An Example Double Check Further Discussion Variation of Parameters For second order equations, we obtain the general formula Z x 1 f (t) Z x 1 f (t) yp(x) = −y1(x) y2(t) dt + y2(x) y1(t) dt, x0 W(y1,y2)(t) a2(t) x0 W(y1,y2)(t) a2(t) where y1, y2 are linearly independent solutions of the 0 0 homogeneous equation and W(y1,y2) := y1y2 − y2y1. W is also called the Wronskian of y1 and y2. logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Variation of Parameters That’s it. Overview An Example Double Check Further Discussion Variation of Parameters For second order equations, we obtain the general formula Z x 1 f (t) Z x 1 f (t) yp(x) = −y1(x) y2(t) dt + y2(x) y1(t) dt, x0 W(y1,y2)(t) a2(t) x0 W(y1,y2)(t) a2(t) where y1, y2 are linearly independent solutions of the 0 0 homogeneous equation and W(y1,y2) := y1y2 − y2y1. W is also called the Wronskian of y1 and y2. The Wronskian can also be represented as y1 y2 W(y1,y2) = det 0 0 . y1 y2 logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Variation of Parameters Overview An Example Double Check Further Discussion Variation of Parameters For second order equations, we obtain the general formula Z x 1 f (t) Z x 1 f (t) yp(x) = −y1(x) y2(t) dt + y2(x) y1(t) dt, x0 W(y1,y2)(t) a2(t) x0 W(y1,y2)(t) a2(t) where y1, y2 are linearly independent solutions of the 0 0 homogeneous equation and W(y1,y2) := y1y2 − y2y1. W is also called the Wronskian of y1 and y2. The Wronskian can also be represented as y1 y2 W(y1,y2) = det 0 0 . y1 y2 That’s it. logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Variation of Parameters Solution of the homogeneous equation. y00 + 4y0 + 4y = 0 λ 2eλx + 4λeλx + 4eλx = 0 2 λ + 4λ + 4 = 0 √ −4 ± 16 − 16 λ = = −2 1,2 2 −2x −2x yh = c1e + c2xe Overview An Example Double Check Further Discussion Solve the Initial Value Problem y00 + 4y0 + 4y = sin(x), y(0) = 1, y0(0) = 0 logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Variation of Parameters y00 + 4y0 + 4y = 0 λ 2eλx + 4λeλx + 4eλx = 0 2 λ + 4λ + 4 = 0 √ −4 ± 16 − 16 λ = = −2 1,2 2 −2x −2x yh = c1e + c2xe Overview An Example Double Check Further Discussion Solve the Initial Value Problem y00 + 4y0 + 4y = sin(x), y(0) = 1, y0(0) = 0 Solution of the homogeneous equation.
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