60.1.86 problem 86

Internal problem ID [10100]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 86
Date solved : Monday, January 27, 2025 at 06:27:28 PM
CAS classification : [[_1st_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.378 (sec). Leaf size: 52

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

\[ \frac {\arctan \left (\frac {\sqrt {a}\, x}{\sqrt {a^{2} y^{2}}}\right )}{\sqrt {a}}-\frac {\left (\int _{}^{y^{2}+\frac {x^{2}}{a}}\frac {f \left (\textit {\_a} a \right )}{\textit {\_a}}d \textit {\_a} \right )}{2}-c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 0.412 (sec). Leaf size: 184

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

\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {-f\left (x^2+a K[2]^2\right ) K[2] a^2-x a}{x^2+a K[2]^2}-\int _1^x\left (\frac {a-2 a^2 K[1] K[2] f''\left (K[1]^2+a K[2]^2\right )}{K[1]^2+a K[2]^2}-\frac {2 a K[2] \left (a K[2]-a f\left (K[1]^2+a K[2]^2\right ) K[1]\right )}{\left (K[1]^2+a K[2]^2\right )^2}\right )dK[1]\right )dK[2]+\int _1^x\frac {a y(x)-a f\left (K[1]^2+a y(x)^2\right ) K[1]}{K[1]^2+a y(x)^2}dK[1]=c_1,y(x)\right ] \]