75.12.27 problem 301

Internal problem ID [16893]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 12. Miscellaneous problems. Exercises page 93
Problem number : 301
Date solved : Tuesday, January 28, 2025 at 09:39:59 AM
CAS classification : [[_homogeneous, `class G`], _rational]

\begin{align*} 2 x^{5}+4 x^{3} y-2 x y^{2}+\left (y^{2}+2 x^{2} y-x^{4}\right ) y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 2.324 (sec). Leaf size: 53

dsolve(2*(x^5+2*x^3*y(x)-y(x)^2*x)+(y(x)^2+2*x^2*y(x)-x^4)*diff(y(x),x)=0,y(x), singsol=all)
 

\begin{align*} y &= \frac {c_{1}}{2}-\frac {\sqrt {-4 x^{4}+4 c_{1} x^{2}+c_{1}^{2}}}{2} \\ y &= \frac {c_{1}}{2}+\frac {\sqrt {-4 x^{4}+4 c_{1} x^{2}+c_{1}^{2}}}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 15.555 (sec). Leaf size: 87

DSolve[2*(x^5+2*x^3*y[x]-y[x]^2*x)+(y[x]^2+2*x^2*y[x]-x^4)*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {1}{2} \left (e^{2 c_1}-\sqrt {-4 x^4+4 e^{2 c_1} x^2+e^{4 c_1}}\right ) \\ y(x)\to \frac {1}{2} \left (\sqrt {-4 x^4+4 e^{2 c_1} x^2+e^{4 c_1}}+e^{2 c_1}\right ) \\ \end{align*}