Solve 1st Order ODE: xyy'=x^2+3y^2, y(1)=2

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Homework Help Overview

The discussion revolves around solving the initial value problem for the first-order ordinary differential equation (ODE) given by xy y' = x^2 + 3y^2 with the initial condition y(1) = 2. Participants are exploring the nature of the equation and the methods for solving it.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • One participant attempts to apply a substitution method, believing the equation to be homogeneous. They express uncertainty regarding their algebra after comparing their results with MATLAB outputs. Another participant confirms the homogeneity and suggests verifying the solution against the original equation and boundary conditions. A further inquiry is made about the differential equation for a transformation involving y^2.

Discussion Status

The discussion is active, with participants providing guidance on checking the validity of the proposed solution. There is an acknowledgment of potential errors in the algebraic manipulation, and multiple interpretations of the problem are being explored.

Contextual Notes

Participants are working under the constraints of the initial value problem and are questioning the correctness of their approaches and assumptions regarding the homogeneity of the differential equation.

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Homework Statement



Find the following IVP Diff.Eq.
xyy'=x^2+3y^2
y(1)=2

Homework Equations


The Attempt at a Solution


I've been struggling with this problem for a while now. I believe I have figured out it is homogenous, thus y=ux substitution applies.
Through some work I have arrived at the answer of
ln x +C=ln(2u^2+1)/4, re-substituting u=y/x I get the answer of
y^2= C*x^3-x^2/2 where C=4.5 based on the IVP.
When I plot this against the ODE solution in MATLAB, the answers do not agree.
Am I wrong that this diff. eqn is homogenous, I believe I have checked my algebra several times. Any help or tips would be great.
 
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You are correct that the equation is homogeneous and that substitution should work. One thing you can try is substitute your solution back into the equation and boundary conditions. If it checks you are home free and if it doesn't, time to check for errors
 
Great, Hopefully my MATLAB code is the problem. I've been trying to find errors in this problem for a while now. Thanks again
 
swtjuice said:
Through some work I have arrived at the answer of
ln x +C=ln(2u^2+1)/4, re-substituting u=y/x I get the answer of
y^2= C*x^3-x^2/2 where C=4.5 based on the IVP.

the blue is correct, but the red is wrong. You made a mistake when removing the logarithm or substituting u=y/x back. Show work in detail.

ehild
 
swtjuice said:

Homework Statement



Find the following IVP Diff.Eq.
xyy'=x^2+3y^2
y(1)=2

Homework Equations





The Attempt at a Solution


I've been struggling with this problem for a while now. I believe I have figured out it is homogenous, thus y=ux substitution applies.
Through some work I have arrived at the answer of
ln x +C=ln(2u^2+1)/4, re-substituting u=y/x I get the answer of
y^2= C*x^3-x^2/2 where C=4.5 based on the IVP.
When I plot this against the ODE solution in MATLAB, the answers do not agree.
Am I wrong that this diff. eqn is homogenous, I believe I have checked my algebra several times. Any help or tips would be great.

What is the DE for z = y^2?

RGV
 

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