60.1.31 problem 31

Internal problem ID [10045]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 31
Date solved : Monday, January 27, 2025 at 06:19:25 PM
CAS classification : [_separable]

\begin{align*} y^{\prime }-a \,x^{n} \left (1+y^{2}\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.016 (sec). Leaf size: 23

dsolve(diff(y(x),x) - a*x^n*(y(x)^2+1)=0,y(x), singsol=all)
 

\[ y = \tan \left (\frac {a \left (x^{n +1}+\left (n +1\right ) c_{1} \right )}{n +1}\right ) \]

✓ Solution by Mathematica

Time used: 0.381 (sec). Leaf size: 52

DSolve[D[y[x],x] - a*x^n*(y[x]^2+1)==0,y[x],x,IncludeSingularSolutions -> True]
 

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