Internal problem ID [8604]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 268.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Bernoulli]
\[ \boxed {f \left (x \right ) y y^{\prime }+g \left (x \right ) y^{2}=-h \left (x \right )} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 114
dsolve(f(x)*y(x)*diff(y(x),x)+g(x)*y(x)^2+h(x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \sqrt {{\mathrm e}^{\int \frac {2 g \left (x \right )}{f \left (x \right )}d x} \left (-2 \left (\int \frac {{\mathrm e}^{\int \frac {2 g \left (x \right )}{f \left (x \right )}d x} h \left (x \right )}{f \left (x \right )}d x \right )+c_{1} \right )}\, {\mathrm e}^{-2 \left (\int \frac {g \left (x \right )}{f \left (x \right )}d x \right )} \\ y \left (x \right ) &= -\sqrt {{\mathrm e}^{\int \frac {2 g \left (x \right )}{f \left (x \right )}d x} \left (-2 \left (\int \frac {{\mathrm e}^{\int \frac {2 g \left (x \right )}{f \left (x \right )}d x} h \left (x \right )}{f \left (x \right )}d x \right )+c_{1} \right )}\, {\mathrm e}^{-2 \left (\int \frac {g \left (x \right )}{f \left (x \right )}d x \right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.329 (sec). Leaf size: 146
DSolve[f[x]*y[x]*y'[x]+g[x]*y[x]^2+h[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\exp \left (\int _1^x-\frac {g(K[1])}{f(K[1])}dK[1]\right ) \sqrt {2 \int _1^x-\frac {\exp \left (-2 \int _1^{K[2]}-\frac {g(K[1])}{f(K[1])}dK[1]\right ) h(K[2])}{f(K[2])}dK[2]+c_1} \\ y(x)\to \exp \left (\int _1^x-\frac {g(K[1])}{f(K[1])}dK[1]\right ) \sqrt {2 \int _1^x-\frac {\exp \left (-2 \int _1^{K[2]}-\frac {g(K[1])}{f(K[1])}dK[1]\right ) h(K[2])}{f(K[2])}dK[2]+c_1} \\ \end{align*}