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Problem #8: Two linearly independent solutions of xy" -(1+x)y' +y = 0 are y₁(x) = 1 + x and y₂(x) = e. (a) Find the Wronskian W(v₁, 2) of y₁ and 12. (b) Using the method of variation of parameters, we want to find a particular solution yp of the following ODE xy" - (1+x)y' +y = x²e²x in the form y(x) = u₁(x)y₁(x) + u₂(x)y₂(x) for some functions u₁ and u₂. The differential equation for u₂(x) can be written as u₂' =.... Type the right-hand side of this equation into the answer box below. (c) Solve the equation from part (b) to find the function 12.

Problem #8: Two linearly independent solutions of xy" -(1+x)y' +y = 0 are y₁(x) = 1 + x and y₂(x) = e. (a) Find the Wronskian W(v₁, 2) of y₁ and 12. (b) Using the method of variation of parameters, we want to find a particular solution yp of the following ODE xy" - (1+x)y' +y = x²e²x in the form y(x) = u₁(x)y₁(x) + u₂(x)y₂(x) for some functions u₁ and u₂. The differential equation for u₂(x) can be written as u₂' =.... Type the right-hand side of this equation into the answer box below. (c) Solve the equation from part (b) to find the function 12.

Advanced Engineering Mathematics
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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Problem #8: Two linearly independent solutions of xy" (1+x)y+y=0 are y₁(x) 1+x and y2(x) = ex. (a) Find the Wronskian Wy₁, y2) of y₁ and y2. (b) Using the method of variation of parameters, we want to find a particular solution yp of the following ODE xy" − (1 + x)y' + y = x²e in the form yp(x) = u(x)y₁(x) + u2(x)y2(x) for some functions ₁ and u₂. The differential equation for u2(x) can be written as u₂' = .... Type the right-hand side of this equation into the answer box below. (c) Solve the equation from part (b) to find the function u2.
Transcribed Image Text:Problem #8: Two linearly independent solutions of xy" (1+x)y+y=0 are y₁(x) 1+x and y2(x) = ex. (a) Find the Wronskian Wy₁, y2) of y₁ and y2. (b) Using the method of variation of parameters, we want to find a particular solution yp of the following ODE xy" − (1 + x)y' + y = x²e in the form yp(x) = u(x)y₁(x) + u2(x)y2(x) for some functions ₁ and u₂. The differential equation for u2(x) can be written as u₂' = .... Type the right-hand side of this equation into the answer box below. (c) Solve the equation from part (b) to find the function u2.
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