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Q5*) Write down an immediate first integral for the Euler-Lagrange equation for the integral I = = F(x, y, y″) dx. Hence write down a first integral of the Euler-Lagrange equation for the integral I 1 = √(xy ² + x³y²) dx. Find the general solution of this ordinary differential equation, seeking first the complementary function and then the particular integral. (Hint: the ODE is of homogeneous degree. And, for the particular integral, try functions proportional to log x.)

Q5*) Write down an immediate first integral for the Euler-Lagrange equation for the integral I = = F(x, y, y″) dx. Hence write down a first integral of the Euler-Lagrange equation for the integral I 1 = √(xy ² + x³y²) dx. Find the general solution of this ordinary differential equation, seeking first the complementary function and then the particular integral. (Hint: the ODE is of homogeneous degree. And, for the particular integral, try functions proportional to log x.)

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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Q5*) Write down an immediate first integral for the Euler-Lagrange equation for the integral I = = F(x, y, y″) dx. Hence write down a first integral of the Euler-Lagrange equation for the integral I 1 = √(xy ² + x³y²) dx. Find the general solution of this ordinary differential equation, seeking first the complementary function and then the particular integral. (Hint: the ODE is of homogeneous degree. And, for the particular integral, try functions proportional to log x.)
Transcribed Image Text:Q5*) Write down an immediate first integral for the Euler-Lagrange equation for the integral I = = F(x, y, y″) dx. Hence write down a first integral of the Euler-Lagrange equation for the integral I 1 = √(xy ² + x³y²) dx. Find the general solution of this ordinary differential equation, seeking first the complementary function and then the particular integral. (Hint: the ODE is of homogeneous degree. And, for the particular integral, try functions proportional to log x.)
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