Internal problem ID [6360]
Book: Own collection of miscellaneous problems
Section: section 1.0
Problem number: 69.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
Solve \begin {gather*} \boxed {a y^{2} y^{\prime \prime }+b y^{2}-c=0} \end {gather*}
✓ Solution by Maple
Time used: 0.187 (sec). Leaf size: 74
dsolve(a*y(x)^2*diff(y(x),x$2)+b*y(x)^2=c,y(x), singsol=all)
\begin{align*} \int _{}^{y \relax (x )}\frac {a \textit {\_a}}{\sqrt {a \textit {\_a} \left (-2 \textit {\_a}^{2} b +\textit {\_a} a c_{1}-2 c \right )}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}-\frac {a \textit {\_a}}{\sqrt {a \textit {\_a} \left (-2 \textit {\_a}^{2} b +\textit {\_a} a c_{1}-2 c \right )}}d \textit {\_a} -x -c_{2} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.176 (sec). Leaf size: 346
DSolve[a*y[x]^2*y''[x]+b*y[x]^2==c,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [-\frac {\left (\sqrt {-16 b c+a^2 c_1{}^2}-a c_1\right ) \left (\sqrt {-16 b c+a^2 c_1{}^2}+a c_1\right ){}^2 \left (1+\frac {4 b y(x)}{\sqrt {-16 b c+a^2 c_1{}^2}-a c_1}\right ) \left (1-\frac {4 b y(x)}{\sqrt {-16 b c+a^2 c_1{}^2}+a c_1}\right ) \left (\text {EllipticE}\left (i \sinh ^{-1}\left (2 \sqrt {y(x)} \sqrt {\frac {b}{\sqrt {-16 b c+a^2 c_1{}^2}-a c_1}}\right ),\frac {a c_1-\sqrt {-16 b c+a^2 c_1{}^2}}{\sqrt {-16 b c+a^2 c_1{}^2}+a c_1}\right )-\text {EllipticF}\left (i \sinh ^{-1}\left (2 \sqrt {y(x)} \sqrt {\frac {b}{\sqrt {-16 b c+a^2 c_1{}^2}-a c_1}}\right ),\frac {a c_1-\sqrt {-16 b c+a^2 c_1{}^2}}{\sqrt {-16 b c+a^2 c_1{}^2}+a c_1}\right )\right ){}^2}{16 b^3 y(x) \left (-\frac {2 \left (b y(x)^2+c\right )}{a y(x)}+c_1\right )}=(x+c_2){}^2,y(x)\right ] \]