60.1.267 problem 268

Internal problem ID [10281]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 268
Date solved : Monday, January 27, 2025 at 06:48:22 PM
CAS classification : [_Bernoulli]

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

✓ Solution by Maple

Time used: 0.023 (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 &= \sqrt {{\mathrm e}^{\int \frac {2 g \left (x \right )}{f}d x} \left (-2 \left (\int \frac {{\mathrm e}^{\int \frac {2 g \left (x \right )}{f}d x} h \left (x \right )}{f}d x \right )+c_{1} \right )}\, {\mathrm e}^{-2 \left (\int \frac {g \left (x \right )}{f}d x \right )} \\ y &= -\sqrt {{\mathrm e}^{\int \frac {2 g \left (x \right )}{f}d x} \left (-2 \left (\int \frac {{\mathrm e}^{\int \frac {2 g \left (x \right )}{f}d x} h \left (x \right )}{f}d x \right )+c_{1} \right )}\, {\mathrm e}^{-2 \left (\int \frac {g \left (x \right )}{f}d x \right )} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.361 (sec). Leaf size: 146

DSolve[f[x]*y[x]*D[y[x],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*}