66.1.47 problem Problem 61

Internal problem ID [13894]
Book : Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS, MOSCOW, Third printing 1977.
Section : Chapter 1, First-Order Differential Equations. Problems page 88
Problem number : Problem 61
Date solved : Tuesday, January 28, 2025 at 06:07:45 AM
CAS classification : [_exact, _rational]

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 316

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

\begin{align*} y &= \frac {\left (-4 x^{3}-12 c_{1} +4 \sqrt {x^{6}+\left (6 c_{1} -4\right ) x^{3}+9 c_{1}^{2}}\right )^{{2}/{3}}+4 x}{2 \left (-4 x^{3}-12 c_{1} +4 \sqrt {x^{6}+\left (6 c_{1} -4\right ) x^{3}+9 c_{1}^{2}}\right )^{{1}/{3}}} \\ y &= \frac {i \left (-\left (-4 x^{3}-12 c_{1} +4 \sqrt {x^{6}+\left (6 c_{1} -4\right ) x^{3}+9 c_{1}^{2}}\right )^{{2}/{3}}+4 x \right ) \sqrt {3}-\left (-4 x^{3}-12 c_{1} +4 \sqrt {x^{6}+\left (6 c_{1} -4\right ) x^{3}+9 c_{1}^{2}}\right )^{{2}/{3}}-4 x}{4 \left (-4 x^{3}-12 c_{1} +4 \sqrt {x^{6}+\left (6 c_{1} -4\right ) x^{3}+9 c_{1}^{2}}\right )^{{1}/{3}}} \\ y &= \frac {i \left (\left (-4 x^{3}-12 c_{1} +4 \sqrt {x^{6}+\left (6 c_{1} -4\right ) x^{3}+9 c_{1}^{2}}\right )^{{2}/{3}}-4 x \right ) \sqrt {3}-\left (-4 x^{3}-12 c_{1} +4 \sqrt {x^{6}+\left (6 c_{1} -4\right ) x^{3}+9 c_{1}^{2}}\right )^{{2}/{3}}-4 x}{4 \left (-4 x^{3}-12 c_{1} +4 \sqrt {x^{6}+\left (6 c_{1} -4\right ) x^{3}+9 c_{1}^{2}}\right )^{{1}/{3}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 3.707 (sec). Leaf size: 326

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

\begin{align*} y(x)\to -\frac {2 x+\sqrt [3]{2} \left (x^3+\sqrt {x^6+(-4+6 c_1) x^3+9 c_1{}^2}+3 c_1\right ){}^{2/3}}{2^{2/3} \sqrt [3]{x^3+\sqrt {x^6+(-4+6 c_1) x^3+9 c_1{}^2}+3 c_1}} \\ y(x)\to \frac {2^{2/3} \left (1-i \sqrt {3}\right ) \left (x^3+\sqrt {x^6+(-4+6 c_1) x^3+9 c_1{}^2}+3 c_1\right ){}^{2/3}+\sqrt [3]{2} \left (2+2 i \sqrt {3}\right ) x}{4 \sqrt [3]{x^3+\sqrt {x^6+(-4+6 c_1) x^3+9 c_1{}^2}+3 c_1}} \\ y(x)\to \frac {2^{2/3} \left (1+i \sqrt {3}\right ) \left (x^3+\sqrt {x^6+(-4+6 c_1) x^3+9 c_1{}^2}+3 c_1\right ){}^{2/3}+\sqrt [3]{2} \left (2-2 i \sqrt {3}\right ) x}{4 \sqrt [3]{x^3+\sqrt {x^6+(-4+6 c_1) x^3+9 c_1{}^2}+3 c_1}} \\ \end{align*}