60.7.182 problem 1773 (book 6.182)

Internal problem ID [11771]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1773 (book 6.182)
Date solved : Tuesday, January 28, 2025 at 06:11:20 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

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

✓ Solution by Maple

Time used: 0.053 (sec). Leaf size: 46

dsolve(x^2*(x-y(x))*diff(diff(y(x),x),x)+a*(x*diff(y(x),x)-y(x))^2=0,y(x), singsol=all)
 

\begin{align*} y &= x \\ y &= -\operatorname {RootOf}\left (-c_{2} a x \,\textit {\_Z}^{a}+c_{1} a \,\textit {\_Z}^{a}+c_{2} x \,\textit {\_Z}^{a}-c_{1} \textit {\_Z}^{a}-x^{a} \textit {\_Z} \right )+x \\ \end{align*}

✓ Solution by Mathematica

Time used: 21.833 (sec). Leaf size: 285

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

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