Internal problem ID [9267]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1688.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]
Solve \begin {gather*} \boxed {x^{4} y^{\prime \prime }-x^{2} \left (x +y^{\prime }\right ) y^{\prime }+4 y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.034 (sec). Leaf size: 32
dsolve(x^4*diff(diff(y(x),x),x)-x^2*(x+diff(y(x),x))*diff(y(x),x)+4*y(x)^2=0,y(x), singsol=all)
\[ y \relax (x ) = \RootOf \left (-\ln \relax (x )+c_{2}-\left (\int _{}^{\textit {\_Z}}\frac {1}{{\mathrm e}^{\textit {\_f}} c_{1}+4 \textit {\_f} +2}d \textit {\_f} \right )\right ) x^{2} \]
✓ Solution by Mathematica
Time used: 0.376 (sec). Leaf size: 189
DSolve[4*y[x]^2 - x^2*y'[x]*(x + y'[x]) + x^4*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\frac {1}{-e^{\frac {K[1]}{x^2}} c_1 x^2+2 x^2+4 K[1]}dK[1]-\int _1^x\left (\frac {K[2] \left (e^{\frac {y(x)}{K[2]^2}} c_1+2 \left (-\frac {y(x)}{K[2]^2}-1\right )\right )}{-e^{\frac {y(x)}{K[2]^2}} c_1 K[2]^2+2 K[2]^2+4 y(x)}+\int _1^{y(x)}-\frac {\frac {2 e^{\frac {K[1]}{K[2]^2}} c_1 K[1]}{K[2]}-2 e^{\frac {K[1]}{K[2]^2}} c_1 K[2]+4 K[2]}{\left (-e^{\frac {K[1]}{K[2]^2}} c_1 K[2]^2+2 K[2]^2+4 K[1]\right ){}^2}dK[1]\right )dK[2]=c_2,y(x)\right ] \]