61.1.3 problem 1.1.3

Internal problem ID [12003]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, First-Order differential equations
Problem number : 1.1.3
Date solved : Monday, January 27, 2025 at 11:48:02 PM
CAS classification : [_separable]

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 20

dsolve(diff(y(x),x)=f(x)*g(y(x)),y(x), singsol=all)
 

\[ \int fd x -\int _{}^{y}\frac {1}{g \left (\textit {\_a} \right )}d \textit {\_a} +c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 0.276 (sec). Leaf size: 42

DSolve[D[y[x],x]==f[x]*g[y[x]],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{g(K[1])}dK[1]\&\right ]\left [\int _1^xf(K[2])dK[2]+c_1\right ] \\ y(x)\to g^{(-1)}(0) \\ \end{align*}