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Solving Nonlinear ODEs I

hatguy

New member
Dec 25, 2012
3
I need to solve 2 ODEs:

1.

2.



but i can't figure out a way to. Please help!
 
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Ackbach

Indicium Physicus
Staff member
Jan 26, 2012
4,197
Regarding 1:

If you let $u=\ln(y)$, then the equation
$$\ln(y)=\left( \frac{y'}{y}\right)^{\!\!2}+x\,\frac{y'}{y}$$
reduces to
$$u=(u')^{2}+xu'.$$
If you view this is a quadratic in $u'$, you can find that
$$u'=\frac{-x \pm \sqrt{x^{2}+4u}}{2}.$$
Not sure where you could go from here. You could try to make it exact.

Regarding 2:

The substitution $u=y^{2}$ renders the equation Ricatti. Have fun with that!
 

Ackbach

Indicium Physicus
Staff member
Jan 26, 2012
4,197
Further update on Number 1: the additional substitution $v=-1+\sqrt{1+4u}$ renders the equation separable, I think.
 

Jester

Well-known member
MHB Math Helper
Jan 26, 2012
183
Q1 Differentiate what Ackbach has giving $u''(2u'+x) = 0$ - two cases to consider.

Q2 As Ackbach said let $u = y^2$, further let $t = x^4$. Your new equation should be homogeneous.
 
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hatguy

New member
Dec 25, 2012
3
Q1 Differentiate what Ackbach has giving $u''(2u'+x) = 0$ - two cases to consider.

Q2 As Ackbach said let $u = y^2$, further let $t = x^4$. Your new equation should be homogeneous.
Thanks, everyone, I found my solutions. And yes, a left bracket shouldn't be where it is now.