60.7.98 problem 1689 (book 6.98)

Internal problem ID [11687]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1689 (book 6.98)
Date solved : Tuesday, January 28, 2025 at 06:07:25 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} x^{4} y^{\prime \prime }-x^{2} \left (x +y^{\prime }\right ) y^{\prime }+4 y^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.063 (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)
 

\[ y = \operatorname {RootOf}\left (-\ln \left (x \right )+c_{2} -\int _{}^{\textit {\_Z}}\frac {1}{{\mathrm e}^{\textit {\_f}} c_{1} +4 \textit {\_f} +2}d \textit {\_f} \right ) x^{2} \]

✓ Solution by Mathematica

Time used: 0.251 (sec). Leaf size: 386

DSolve[4*y[x]^2 - x^2*D[y[x],x]*(x + D[y[x],x]) + x^4*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^{y(x)}\frac {1}{-e^{\frac {K[2]}{x^2}} c_1 x^2-e^{\frac {K[2]}{x^2}} \int _1^{\frac {K[2]}{x^2}}2 e^{-K[1]} K[1]dK[1] x^2+2 K[2]}dK[2]-\int _1^x\left (\frac {K[3] \left (e^{\frac {y(x)}{K[3]^2}} c_1+e^{\frac {y(x)}{K[3]^2}} \int _1^{\frac {y(x)}{K[3]^2}}2 e^{-K[1]} K[1]dK[1]\right )}{-e^{\frac {y(x)}{K[3]^2}} c_1 K[3]^2-e^{\frac {y(x)}{K[3]^2}} \int _1^{\frac {y(x)}{K[3]^2}}2 e^{-K[1]} K[1]dK[1] K[3]^2+2 y(x)}+\int _1^{y(x)}-\frac {\frac {4 K[2]^2}{K[3]^3}+\frac {2 e^{\frac {K[2]}{K[3]^2}} \int _1^{\frac {K[2]}{K[3]^2}}2 e^{-K[1]} K[1]dK[1] K[2]}{K[3]}+\frac {2 e^{\frac {K[2]}{K[3]^2}} c_1 K[2]}{K[3]}-2 e^{\frac {K[2]}{K[3]^2}} c_1 K[3]-2 e^{\frac {K[2]}{K[3]^2}} K[3] \int _1^{\frac {K[2]}{K[3]^2}}2 e^{-K[1]} K[1]dK[1]}{\left (-e^{\frac {K[2]}{K[3]^2}} c_1 K[3]^2-e^{\frac {K[2]}{K[3]^2}} \int _1^{\frac {K[2]}{K[3]^2}}2 e^{-K[1]} K[1]dK[1] K[3]^2+2 K[2]\right ){}^2}dK[2]\right )dK[3]=c_2,y(x)\right ] \]