60.2.173 problem 749

Internal problem ID [10760]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 749
Date solved : Tuesday, January 28, 2025 at 05:11:44 PM
CAS classification : [_rational]

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

Solution by Maple

Time used: 0.014 (sec). Leaf size: 186

dsolve(diff(y(x),x) = (x-y(x))^2*(x+y(x))^2*x/y(x),y(x), singsol=all)
 
\begin{align*} y &= \frac {\sqrt {\left (c_{1} \left (x^{2}+1\right ) {\mathrm e}^{-\frac {\left (x^{2}+1\right )^{2}}{2}}+\left (x^{2}-1\right ) {\mathrm e}^{-\frac {x^{2} \left (x^{2}-2\right )}{2}}\right ) \left (c_{1} {\mathrm e}^{-\frac {\left (x^{2}+1\right )^{2}}{2}}+{\mathrm e}^{-\frac {x^{2} \left (x^{2}-2\right )}{2}}\right )}}{c_{1} {\mathrm e}^{-\frac {\left (x^{2}+1\right )^{2}}{2}}+{\mathrm e}^{-\frac {x^{2} \left (x^{2}-2\right )}{2}}} \\ y &= -\frac {\sqrt {\left (c_{1} \left (x^{2}+1\right ) {\mathrm e}^{-\frac {\left (x^{2}+1\right )^{2}}{2}}+\left (x^{2}-1\right ) {\mathrm e}^{-\frac {x^{2} \left (x^{2}-2\right )}{2}}\right ) \left (c_{1} {\mathrm e}^{-\frac {\left (x^{2}+1\right )^{2}}{2}}+{\mathrm e}^{-\frac {x^{2} \left (x^{2}-2\right )}{2}}\right )}}{c_{1} {\mathrm e}^{-\frac {\left (x^{2}+1\right )^{2}}{2}}+{\mathrm e}^{-\frac {x^{2} \left (x^{2}-2\right )}{2}}} \\ \end{align*}

Solution by Mathematica

Time used: 0.161 (sec). Leaf size: 155

DSolve[D[y[x],x] == (x*(x - y[x])^2*(x + y[x])^2)/y[x],y[x],x,IncludeSingularSolutions -> True]
 
\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {K[2]}{2 \left (-x^2+K[2]^2-1\right )}-\frac {K[2]}{2 \left (-x^2+K[2]^2+1\right )}-\int _1^x\left (\frac {K[1] K[2]}{\left (K[1]^2-K[2]^2+1\right )^2}-\frac {K[1] K[2]}{\left (K[1]^2-K[2]^2-1\right )^2}\right )dK[1]\right )dK[2]+\int _1^x\left (-\frac {K[1]}{2 \left (K[1]^2-y(x)^2-1\right )}+\frac {K[1]}{2 \left (K[1]^2-y(x)^2+1\right )}-K[1]\right )dK[1]=c_1,y(x)\right ] \]