Problem #5 Solving 2nd-Order, Linear, Homogeneous ODES Given the 2nd-order, linear, homogeneous ODE y"(x) + P(x)y'(x) + Q(x)y(x) = 0, it can be shown that if the quantity Q'x) + 2P(x)Q(x) (Q(x))32 is a constant, then the transformation -S JaQx) dx = Z for any non-zero constant a will transform the original ODE to the ODE Y"(2) + -Y'(2) + 금y(2) = 0 2 Ja which has constant coefficients. The solution to the original ODE is then - [ JaQ¢x) dx. y(x) = ¥(2) with Z = Use this idea to determine a general solution to the following ODE 2xy" (x) + y'(x) + dy(x) = 0 for x > 0.
Problem #5 Solving 2nd-Order, Linear, Homogeneous ODES Given the 2nd-order, linear, homogeneous ODE y"(x) + P(x)y'(x) + Q(x)y(x) = 0, it can be shown that if the quantity Q'x) + 2P(x)Q(x) (Q(x))32 is a constant, then the transformation -S JaQx) dx = Z for any non-zero constant a will transform the original ODE to the ODE Y"(2) + -Y'(2) + 금y(2) = 0 2 Ja which has constant coefficients. The solution to the original ODE is then - [ JaQ¢x) dx. y(x) = ¥(2) with Z = Use this idea to determine a general solution to the following ODE 2xy" (x) + y'(x) + dy(x) = 0 for x > 0.
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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The second image is problem 5 and its solution (hint for the question provided)
![**Another Regular Sturm-Liouville Problem**
Determine the eigenvalues (\(\lambda_n\)) and eigenfunctions (\(\varphi_n(x)\)) for the differential equation
\[ 2x \varphi''(x) + \varphi'(x) + \lambda \varphi(x) = 0 \]
for \( 0 \leq x \leq 1 \), along with the boundary conditions, \(\varphi(0) = \varphi(1) = 0\). *Hint: See Problem #5 of Homework #4.*
Determine the "dot" product for which \(\varphi_m \cdot \varphi_n = 0\) when \(m \neq n\) and use this to determine the coefficients \(a_n\) if the function \(f(x) = x\) is expanded as
\[ f(x) = x = \sum_{n=1}^{\infty} a_n \varphi_n(x) \]
for \(0 \leq x \leq 1\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe5d528f6-f704-4253-8470-44f2fbc0de86%2F5eccb21f-7ee2-496c-bb64-335920cd19fe%2Fe1g27m_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Another Regular Sturm-Liouville Problem**
Determine the eigenvalues (\(\lambda_n\)) and eigenfunctions (\(\varphi_n(x)\)) for the differential equation
\[ 2x \varphi''(x) + \varphi'(x) + \lambda \varphi(x) = 0 \]
for \( 0 \leq x \leq 1 \), along with the boundary conditions, \(\varphi(0) = \varphi(1) = 0\). *Hint: See Problem #5 of Homework #4.*
Determine the "dot" product for which \(\varphi_m \cdot \varphi_n = 0\) when \(m \neq n\) and use this to determine the coefficients \(a_n\) if the function \(f(x) = x\) is expanded as
\[ f(x) = x = \sum_{n=1}^{\infty} a_n \varphi_n(x) \]
for \(0 \leq x \leq 1\).
![**Problem #5: Solving 2nd-Order, Linear, Homogeneous ODEs**
Given the 2nd-order, linear, homogeneous ordinary differential equation (ODE):
\[ y''(x) + P(x)y'(x) + Q(x)y(x) = 0, \]
it can be shown that if the quantity
\[ \gamma = \frac{Q'(x) + 2P(x)Q(x)}{(Q(x))^{3/2}} \]
is a constant, then the transformation
\[ z = \int \sqrt{\alpha Q(x)} \, dx \]
for any non-zero constant \( \alpha \) will transform the original ODE to the ODE
\[ \Psi''(z) + \frac{\gamma}{2\sqrt{\alpha}} \Psi'(z) + \frac{1}{\alpha} \Psi(z) = 0 \]
which has constant coefficients. The solution to the original ODE is then
\[ y(x) = \Psi(z) \quad \text{with} \quad z = \int \sqrt{\alpha Q(x)} \, dx. \]
Use this idea to determine a general solution to the following ODE:
\[ 2xy''(x) + y'(x) + \lambda y(x) = 0 \]
for \( x > 0 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe5d528f6-f704-4253-8470-44f2fbc0de86%2F5eccb21f-7ee2-496c-bb64-335920cd19fe%2Fwbeg7px_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem #5: Solving 2nd-Order, Linear, Homogeneous ODEs**
Given the 2nd-order, linear, homogeneous ordinary differential equation (ODE):
\[ y''(x) + P(x)y'(x) + Q(x)y(x) = 0, \]
it can be shown that if the quantity
\[ \gamma = \frac{Q'(x) + 2P(x)Q(x)}{(Q(x))^{3/2}} \]
is a constant, then the transformation
\[ z = \int \sqrt{\alpha Q(x)} \, dx \]
for any non-zero constant \( \alpha \) will transform the original ODE to the ODE
\[ \Psi''(z) + \frac{\gamma}{2\sqrt{\alpha}} \Psi'(z) + \frac{1}{\alpha} \Psi(z) = 0 \]
which has constant coefficients. The solution to the original ODE is then
\[ y(x) = \Psi(z) \quad \text{with} \quad z = \int \sqrt{\alpha Q(x)} \, dx. \]
Use this idea to determine a general solution to the following ODE:
\[ 2xy''(x) + y'(x) + \lambda y(x) = 0 \]
for \( x > 0 \).
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