Solve the following differential equations: $2 x^{2} y \frac{d^{2} y}{{d x}^{2}} + 4 y^{2} = x^{2} {(\frac{d y}{d x})}^{2} + 2 x y (\frac{d y}{d x})$ $2x^(2)y (d^(2)y)/(dx^(2))+4y^(2)=x^(2)(dy/dx)^(2)+2xy(dy/dx)$ ?

Question

3354 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-08-11T21:02:24

$y = x^{2} {(C_{1} + C_{2} ln x)}_{1}^{2}$ $y = x^2(C_1+C_2 ln x)^2$

Making the change of variables $y (x) = x^{2} z^{2}$ $y(x) = x^2 z^2$ and substituting into the DE we get

$4x^5 z^3(z'+x z'') = 0$

so assuming $z \ne 0$ we have

$z'+xz'' = 0\to z = C_1+C_2 ln x$

and then

$y = x^2(C_1+C_2 ln x)^2$

Solve the following differential equations: 2x^(2)y (d^(2)y)/(dx^(2))+4y^(2)=x^(2)(dy/dx)^(2)+2xy(dy/dx)2x2yd2ydx2+4y2=x2(dydx)2+2xy(dydx)2x^(2)y (d^(2)y)/(dx^(2))+4y^(2)=x^(2)(dy/dx)^(2)+2xy(dy/dx)?