Nonlinear differential equations of the second order – need your help

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November 27th 2010, 05:07 AM
Metrika

Nonlinear differential equations of the second order – need your help

Please help me solve these two nonlinear differential equations of the second order

1) $x^3y''=(y-xy')(y-xy'-x)$

2) $x^2(yy''-(y')^2)+xyy'=(2xy'-3y)\sqrt{x^3}$

I do not even know how to begin (Blush)
November 27th 2010, 07:41 AM
DeMath

Quote:

Originally Posted by Metrika

Please help me solve these two nonlinear differential equations of the second order

1) $x^3y''=(y-xy')(y-xy'-x)$

$\displaystyle{x^3y''=(y-xy')(y-xy'-x)~\Rightarrow~x^3y''=(xy'-y)^2+x(xy'-y)~\Rightarrow}$

$\displaystyle{\Rightarrow~y''=x\!\left(\frac{xy'-y}{x^2}\right)^2+\frac{xy'-y}{x^2}~\Rightarrow~y''=x\!\left(\frac{y}{x}\right )'^2+\left(\frac{y}{x}\right)'\Rightarrow}$

$\displaystyle{\Rightarrow~\!\left\{\begin{gathered }y=xz\hfill\\y'=z+xz'\hfill\\y''=2z'+xz''\hfill\\\ end{gathered}\right\}\!~\Rightarrow~xz''-xz'^2+z'=0}$

Consider two cases:

1) $z'=0~\Rightarrow~z=C~\Rightarrow~y=Cx$ ;

2) $z'\ne0$ :

$\displaystyle{\frac{z''}{z'}-z'+\frac{1}{x}=0~\Rightarrow~\left(\ln{z'}-z+\ln{x}\right)^\prime}=0~\Rightarrow~\ln|z'|-z+\ln|x|=C_1~\Rightarrow}$

$\displaystyle{\Rightarrow~\ln|xz'|=C_1+z~\Rightarr ow~xz'=e^{C_1+z}~\Rightarrow~\int{e^{-(C_1+z)}\,dz}=\int\frac{dx}{x}~\Rightarrow}$

$\displaystyle{\Rightarrow~-e^{-(C_1+z)}=\ln|x|+C_2~\Rightarrow~e^{-(C_1+z)}=\ln\left|\frac{1}{C_2x}\right|\Rightarrow ~-(C_1+z)=\ln\ln\left|\frac{1}{C_2x}\right|\Rightarr ow}$

$\displaystyle{\Rightarrow~z=-C_1-\ln\ln\left|\frac{1}{C_2x}\right|=\ln\frac{C_3}{\l n|C_2x|}~\Rightarrow~y=x\ln\frac{C_3}{\ln|C_2x|}}$
November 27th 2010, 10:08 AM
Krizalid

Quote:

Originally Posted by Metrika

Please help me solve these two nonlinear differential equations of the second order

1) $x^3y''=(y-xy')(y-xy'-x)$

substitute $t=y-xy'$ and you'll end up with a homogeneous DE, then put $t=ux$ and the reamining equation will be linear, everything straightforward and you'll get the result.
November 27th 2010, 10:16 AM
Danny

For the second, divide by $xy^2$ and then re-write as

$x \dfrac{d}{dx} \left( \dfrac{y'}{y}\right) + \dfrac{y'}{y} = 2 \left( \dfrac{x^{3/2}}{y}\right)^2 \dfrac{d}{dx} \left(\dfrac{y}{x^{3/2}} \right)$