29.21.21 problem 597

Internal problem ID [5191]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 21
Problem number : 597
Date solved : Monday, January 27, 2025 at 10:18:39 AM
CAS classification : [_exact, _rational]

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

✓ Solution by Maple

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

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

\begin{align*} y \left (x \right ) &= \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 \left (x \right ) &= \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 \left (x \right ) &= \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: 3.957 (sec). Leaf size: 326

DSolve[(x-y[x]^2)D[y[x],x]==x^2-y[x],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*}