29.22.6 problem 612

Internal problem ID [5206]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 22
Problem number : 612
Date solved : Monday, January 27, 2025 at 10:20:56 AM
CAS classification : [_exact, _rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 442

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

\begin{align*} y \left (x \right ) &= \frac {\left (12 x^{3}+12 c_{1} +8+4 \sqrt {-4 x^{9}-3 x^{6}+18 c_{1} x^{3}+9 c_{1}^{2}+12 c_{1}}\right )^{{1}/{3}}}{2}+\frac {2 x^{3}+2}{\left (12 x^{3}+12 c_{1} +8+4 \sqrt {-4 x^{9}-3 x^{6}+18 c_{1} x^{3}+9 c_{1}^{2}+12 c_{1}}\right )^{{1}/{3}}}+1 \\ y \left (x \right ) &= -\frac {\left (12 x^{3}+12 c_{1} +8+4 \sqrt {-4 x^{9}-3 x^{6}+18 c_{1} x^{3}+9 c_{1}^{2}+12 c_{1}}\right )^{{1}/{3}}}{4}+\frac {-x^{3}-1}{\left (12 x^{3}+12 c_{1} +8+4 \sqrt {-4 x^{9}-3 x^{6}+18 c_{1} x^{3}+9 c_{1}^{2}+12 c_{1}}\right )^{{1}/{3}}}+1+\frac {i \left (x^{3}-\frac {\left (12 x^{3}+12 c_{1} +8+4 \sqrt {-4 x^{9}-3 x^{6}+18 c_{1} x^{3}+9 c_{1}^{2}+12 c_{1}}\right )^{{2}/{3}}}{4}+1\right ) \sqrt {3}}{\left (12 x^{3}+12 c_{1} +8+4 \sqrt {-4 x^{9}-3 x^{6}+18 c_{1} x^{3}+9 c_{1}^{2}+12 c_{1}}\right )^{{1}/{3}}} \\ y \left (x \right ) &= \frac {\frac {\left (i \sqrt {3}-1\right ) \left (12 x^{3}+12 c_{1} +8+4 \sqrt {-4 x^{9}-3 x^{6}+18 c_{1} x^{3}+9 c_{1}^{2}+12 c_{1}}\right )^{{2}/{3}}}{4}-1+\left (12 x^{3}+12 c_{1} +8+4 \sqrt {-4 x^{9}-3 x^{6}+18 c_{1} x^{3}+9 c_{1}^{2}+12 c_{1}}\right )^{{1}/{3}}-i \left (x^{3}+1\right ) \sqrt {3}-x^{3}}{\left (12 x^{3}+12 c_{1} +8+4 \sqrt {-4 x^{9}-3 x^{6}+18 c_{1} x^{3}+9 c_{1}^{2}+12 c_{1}}\right )^{{1}/{3}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 5.512 (sec). Leaf size: 409

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

\begin{align*} y(x)\to -\frac {\sqrt [3]{2} \left (x^3+1\right )}{\sqrt [3]{-3 x^3+\sqrt {-4 x^9-3 x^6-18 c_1 x^3+3 c_1 (-4+3 c_1)}-2+3 c_1}}-\frac {\sqrt [3]{-3 x^3+\sqrt {-4 x^9-3 x^6-18 c_1 x^3+3 c_1 (-4+3 c_1)}-2+3 c_1}}{\sqrt [3]{2}}+1 \\ y(x)\to \frac {\left (1+i \sqrt {3}\right ) \left (x^3+1\right )}{2^{2/3} \sqrt [3]{-3 x^3+\sqrt {-4 x^9-3 x^6-18 c_1 x^3+3 c_1 (-4+3 c_1)}-2+3 c_1}}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{-3 x^3+\sqrt {-4 x^9-3 x^6-18 c_1 x^3+3 c_1 (-4+3 c_1)}-2+3 c_1}}{2 \sqrt [3]{2}}+1 \\ y(x)\to \frac {\left (1-i \sqrt {3}\right ) \left (x^3+1\right )}{2^{2/3} \sqrt [3]{-3 x^3+\sqrt {-4 x^9-3 x^6-18 c_1 x^3+3 c_1 (-4+3 c_1)}-2+3 c_1}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-3 x^3+\sqrt {-4 x^9-3 x^6-18 c_1 x^3+3 c_1 (-4+3 c_1)}-2+3 c_1}}{2 \sqrt [3]{2}}+1 \\ y(x)\to 0 \\ \end{align*}