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

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The forum discussion focuses on solving the first-order ordinary differential equation (ODE) given by xy y' = x² + 3y² with the initial value problem (IVP) y(1) = 2. The user identifies the equation as homogeneous and applies the substitution y = ux, leading to the derived solution ln x + C = ln(2u² + 1)/4, resulting in y² = Cx³ - x²/2 with C = 4.5. However, discrepancies arise when comparing this solution to MATLAB outputs, prompting a request for verification of the solution process.

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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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