60.1.229 problem 230

Internal problem ID [10243]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 230
Date solved : Monday, January 27, 2025 at 06:41:12 PM
CAS classification : [_Bernoulli]

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

✓ Solution by Maple

Time used: 0.028 (sec). Leaf size: 98

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

\begin{align*} y &= \frac {\sqrt {a \,{\mathrm e}^{\frac {2 b x}{a}} \left (c_{1} a -2 \left (\int {\mathrm e}^{\frac {2 b x}{a}} fd x \right )\right )}\, {\mathrm e}^{-\frac {2 b x}{a}}}{a} \\ y &= -\frac {\sqrt {a \,{\mathrm e}^{\frac {2 b x}{a}} \left (c_{1} a -2 \left (\int {\mathrm e}^{\frac {2 b x}{a}} fd x \right )\right )}\, {\mathrm e}^{-\frac {2 b x}{a}}}{a} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.355 (sec). Leaf size: 98

DSolve[a*y[x]*D[y[x],x]+b*y[x]^2+f[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -e^{-\frac {b x}{a}} \sqrt {2 \int _1^x-\frac {e^{\frac {2 b K[1]}{a}} f(K[1])}{a}dK[1]+c_1} \\ y(x)\to e^{-\frac {b x}{a}} \sqrt {2 \int _1^x-\frac {e^{\frac {2 b K[1]}{a}} f(K[1])}{a}dK[1]+c_1} \\ \end{align*}