Internal problem ID [7113]
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]]
\[ \boxed {a y^{2} y^{\prime \prime }+b y^{2}=c} \]
✓ Solution by Maple
Time used: 0.109 (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 \left (x \right )}\frac {\textit {\_a} a}{\sqrt {\textit {\_a} a \left (c_{1} \textit {\_a} a -2 b \,\textit {\_a}^{2}-2 c \right )}}d \textit {\_a} -x -c_{2} = 0 \int _{}^{y \left (x \right )}-\frac {\textit {\_a} a}{\sqrt {\textit {\_a} a \left (c_{1} \textit {\_a} a -2 b \,\textit {\_a}^{2}-2 c \right )}}d \textit {\_a} -x -c_{2} = 0 \end{align*}
✓ Solution by Mathematica
Time used: 0.801 (sec). Leaf size: 346
DSolve[a*y[x]^2*y''[x]+b*y[x]^2==c,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [-\frac {\left (\sqrt {-16 b c+a^2 c_1{}^2}-a c_1\right ) \left (\sqrt {-16 b c+a^2 c_1{}^2}+a c_1\right ){}^2 \left (1+\frac {4 b y(x)}{\sqrt {-16 b c+a^2 c_1{}^2}-a c_1}\right ) \left (1-\frac {4 b y(x)}{\sqrt {-16 b c+a^2 c_1{}^2}+a c_1}\right ) \left (E\left (i \text {arcsinh}\left (2 \sqrt {\frac {b}{\sqrt {a^2 c_1{}^2-16 b c}-a c_1}} \sqrt {y(x)}\right )|\frac {a c_1-\sqrt {a^2 c_1{}^2-16 b c}}{a c_1+\sqrt {a^2 c_1{}^2-16 b c}}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (2 \sqrt {\frac {b}{\sqrt {a^2 c_1{}^2-16 b c}-a c_1}} \sqrt {y(x)}\right ),\frac {a c_1-\sqrt {a^2 c_1{}^2-16 b c}}{a c_1+\sqrt {a^2 c_1{}^2-16 b c}}\right )\right ){}^2}{16 b^3 y(x) \left (-\frac {2 \left (b y(x)^2+c\right )}{a y(x)}+c_1\right )}=(x+c_2){}^2,y(x)\right ] \]