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1. Follow the procedure above to solve the nonlinear ODE x²y + 2xy-y³=0, x>0. x² y' + 2xy = y³ =0, x>0. - 2. Find the solution, given the initial condition y(1) = 1.

1. Follow the procedure above to solve the nonlinear ODE x²y + 2xy-y³=0, x>0. x² y' + 2xy = y³ =0, x>0. - 2. Find the solution, given the initial condition y(1) = 1.

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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Question

bernoulli differential equation

They often appear in the study of population dynamics and fluids. Form=0,1thesubstitutionv=y¹n reduces equation (1) to a linear equation, makingit amenable dy dx + p(x) y = q(x) y", nЄR. to the integrating factor technique. From equation (1), dividing through by y" yields dy dx Now u = 1-n dv d.x Equation (2) can now be written as the linear equation, 1 dv n dx 1 = (1 − n) y-n -n y + p(x) y¹−n = q(x). dy dx + p(x) v = q (x). 2. Find the solution, given the initial condition y(1) = 1. -n ⇒ y dy 1 dx 1- = With equation (3) solved for v the original substitution can be recalled to solve for y. 1. Follow the procedure above to solve the nonlinear ODE x²y + 2xy-y³=0, x>0. x² y² + 2xy = y³ = 0, - x > 0. dv n dx (3)
Transcribed Image Text:They often appear in the study of population dynamics and fluids. Form=0,1thesubstitutionv=y¹n reduces equation (1) to a linear equation, makingit amenable dy dx + p(x) y = q(x) y", nЄR. to the integrating factor technique. From equation (1), dividing through by y" yields dy dx Now u = 1-n dv d.x Equation (2) can now be written as the linear equation, 1 dv n dx 1 = (1 − n) y-n -n y + p(x) y¹−n = q(x). dy dx + p(x) v = q (x). 2. Find the solution, given the initial condition y(1) = 1. -n ⇒ y dy 1 dx 1- = With equation (3) solved for v the original substitution can be recalled to solve for y. 1. Follow the procedure above to solve the nonlinear ODE x²y + 2xy-y³=0, x>0. x² y² + 2xy = y³ = 0, - x > 0. dv n dx (3)
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