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Let u = Y, or y = ux. Then dy = u dx + x du by the product rule. Use this substitution to simplify the given equation. xy2 dy + (-y3 + x3) dx = 0 xy2 dy – y3 dx + x3 dx = их (u dx + x du) – (ux)3 dx + x³ dx = 0 xy? xy- + dx + u2x4 du – (ux)³ dx + x³ dx dx + 2 du

Let u = Y, or y = ux. Then dy = u dx + x du by the product rule. Use this substitution to simplify the given equation. xy2 dy + (-y3 + x3) dx = 0 xy2 dy – y3 dx + x3 dx = их (u dx + x du) – (ux)3 dx + x³ dx = 0 xy? xy- + dx + u2x4 du – (ux)³ dx + x³ dx dx + 2 du

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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The strategy to find a solution to a homogenous differential equation is to make a substitution that results in a separable equation. For a homogenous equation, either of the substitutions u Y or v = 4. will make this y work. Let u = Y, or y = ux. Then dy = u dx + x du by the product rule. Use this substitution to simplify the given equation. xy2 dy + (-y3 + x3) dx = 0 xy2 dy – y3 dx + x³ dx = 0 их (u dx + x du) - (ux)3 dx + x3 dx 2 xy + dx + u?x4 du - (ux)3 dx + x³ dx = 0 dx + du = 0
Transcribed Image Text:The strategy to find a solution to a homogenous differential equation is to make a substitution that results in a separable equation. For a homogenous equation, either of the substitutions u Y or v = 4. will make this y work. Let u = Y, or y = ux. Then dy = u dx + x du by the product rule. Use this substitution to simplify the given equation. xy2 dy + (-y3 + x3) dx = 0 xy2 dy – y3 dx + x³ dx = 0 их (u dx + x du) - (ux)3 dx + x3 dx 2 xy + dx + u?x4 du - (ux)3 dx + x³ dx = 0 dx + du = 0
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Step 1

Given:

xy2dy+-y3+x3dx=0xy2dy-y3dx+x3dx=0    . . .(1)

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