29.21.7 problem 583

Internal problem ID [5177]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 21
Problem number : 583
Date solved : Monday, January 27, 2025 at 10:17:55 AM
CAS classification : [[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} \left (1-x^{3} y\right ) y^{\prime }&=x^{2} y^{2} \end{align*}

✓ Solution by Maple

Time used: 0.919 (sec). Leaf size: 673

dsolve((1-x^3*y(x))*diff(y(x),x) = x^2*y(x)^2,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 50.735 (sec). Leaf size: 331

DSolve[(1-x^3 y[x])D[y[x],x]==x^2 y[x]^2,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {\sqrt [3]{12 c_1 x^6+2 \sqrt {6} \sqrt {c_1 x^6 \left (1+6 c_1 x^6\right )}+1}+\frac {1}{\sqrt [3]{12 c_1 x^6+2 \sqrt {6} \sqrt {c_1 x^6 \left (1+6 c_1 x^6\right )}+1}}+1}{2 x^3} \\ y(x)\to \frac {2 i \left (\sqrt {3}+i\right ) \sqrt [3]{12 c_1 x^6+2 \sqrt {6} \sqrt {c_1 x^6 \left (1+6 c_1 x^6\right )}+1}-\frac {2 \left (1+i \sqrt {3}\right )}{\sqrt [3]{12 c_1 x^6+2 \sqrt {6} \sqrt {c_1 x^6 \left (1+6 c_1 x^6\right )}+1}}+4}{8 x^3} \\ y(x)\to \frac {-2 \left (1+i \sqrt {3}\right ) \sqrt [3]{12 c_1 x^6+2 \sqrt {6} \sqrt {c_1 x^6 \left (1+6 c_1 x^6\right )}+1}+\frac {2 i \left (\sqrt {3}+i\right )}{\sqrt [3]{12 c_1 x^6+2 \sqrt {6} \sqrt {c_1 x^6 \left (1+6 c_1 x^6\right )}+1}}+4}{8 x^3} \\ y(x)\to 0 \\ \end{align*}