For the differential equation (1 − 2x)y'' + 2y' + (2x − 3)y = e−x(a) Write the associated homogeneous equation and check that y1(x) = e x and y2(x) = xe−x are two linearly independent solutions of the associated equations.(b) Check that yp(x) = − 1/4 e−x is a solution of the full equation.(c) Use superposition principle to write down the general solution of the full equation.(d) Use the existence theorem or the Wronskian to find the intervals on which the solution is valid. 7. For the differential equation (12x)y" + 2y' + (2x - 3)y = e(a) Write the associated homogeneous equation and check that y₁(x) = eª and y₂(x) = xe are twolinearly independent solutions of the associated equations.(b) Check that yp(x) = -e-² is a solution of the full equation.(c) Use superposition principle to write down the general solution of the full equation.(d) Use the existence theorem or the Wronskian to find the intervals on which the solution is valid.

Answered: Image /qna-images/answer/ce570cb2-f773-4ab4-8a21-50aa17d6f994.jpg

7. For the differential equation (12x)y" + 2y' + (2x - 3)y=e=" (a) Write the associated homogeneous equation and check that y₁(x) = e and y₂(x) = xe are linearly independent solutions of the associated equations. (b) Check that yp(x) = −e-ª is a solution of the full equation. (c) Use superposition principle to write down the general solution of the full equation.

7. For the differential equation (12x)y" + 2y' + (2x - 3)y=e=" (a) Write the associated homogeneous equation and check that y₁(x) = e and y₂(x) = xe are linearly independent solutions of the associated equations. (b) Check that yp(x) = −e-ª is a solution of the full equation. (c) Use superposition principle to write down the general solution of the full equation.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

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For the differential equation (1 − 2x)y'' + 2y' + (2x − 3)y = e^−x

(a) Write the associated homogeneous equation and check that y₁(x) = e x and y₂(x) = xe^−x are two linearly independent solutions of the associated equations.

(b) Check that y_p(x) = − 1/4 e^−x is a solution of the full equation.

(d) Use the existence theorem or the Wronskian to find the intervals on which the solution is valid.