29.22.3 problem 609
Internal
problem
ID
[5203]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
22
Problem
number
:
609
Date
solved
:
Monday, January 27, 2025 at 10:20:48 AM
CAS
classification
:
[_separable]
\begin{align*} y \left (y+1\right ) y^{\prime }&=\left (1+x \right ) x \end{align*}
✓ Solution by Maple
Time used: 0.004 (sec). Leaf size: 494
dsolve(y(x)*(1+y(x))*diff(y(x),x) = x*(1+x),y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= \frac {\left (-1+4 x^{3}+6 x^{2}+12 c_{1} +2 \sqrt {4 x^{6}+12 x^{5}+24 c_{1} x^{3}+9 x^{4}+36 c_{1} x^{2}-2 x^{3}+36 c_{1}^{2}-3 x^{2}-6 c_{1}}\right )^{{1}/{3}}}{2}+\frac {1}{2 \left (-1+4 x^{3}+6 x^{2}+12 c_{1} +2 \sqrt {4 x^{6}+12 x^{5}+24 c_{1} x^{3}+9 x^{4}+36 c_{1} x^{2}-2 x^{3}+36 c_{1}^{2}-3 x^{2}-6 c_{1}}\right )^{{1}/{3}}}-\frac {1}{2} \\
y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) \left (4 x^{3}+6 x^{2}+2 \sqrt {\left (2 x^{3}+3 x^{2}+6 c_{1} \right ) \left (2 x^{3}+3 x^{2}+6 c_{1} -1\right )}+12 c_{1} -1\right )^{{2}/{3}}-i \sqrt {3}+2 \left (4 x^{3}+6 x^{2}+2 \sqrt {\left (2 x^{3}+3 x^{2}+6 c_{1} \right ) \left (2 x^{3}+3 x^{2}+6 c_{1} -1\right )}+12 c_{1} -1\right )^{{1}/{3}}+1}{4 \left (4 x^{3}+6 x^{2}+2 \sqrt {\left (2 x^{3}+3 x^{2}+6 c_{1} \right ) \left (2 x^{3}+3 x^{2}+6 c_{1} -1\right )}+12 c_{1} -1\right )^{{1}/{3}}} \\
y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) \left (4 x^{3}+6 x^{2}+2 \sqrt {\left (2 x^{3}+3 x^{2}+6 c_{1} \right ) \left (2 x^{3}+3 x^{2}+6 c_{1} -1\right )}+12 c_{1} -1\right )^{{2}/{3}}-i \sqrt {3}-2 \left (4 x^{3}+6 x^{2}+2 \sqrt {\left (2 x^{3}+3 x^{2}+6 c_{1} \right ) \left (2 x^{3}+3 x^{2}+6 c_{1} -1\right )}+12 c_{1} -1\right )^{{1}/{3}}-1}{4 \left (4 x^{3}+6 x^{2}+2 \sqrt {\left (2 x^{3}+3 x^{2}+6 c_{1} \right ) \left (2 x^{3}+3 x^{2}+6 c_{1} -1\right )}+12 c_{1} -1\right )^{{1}/{3}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 4.352 (sec). Leaf size: 346
DSolve[y[x]*(1+y[x])*D[y[x],x]==x*(1+x),y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {1}{2} \left (\sqrt [3]{4 x^3+6 x^2+\sqrt {-1+\left (4 x^3+6 x^2-1+12 c_1\right ){}^2}-1+12 c_1}+\frac {1}{\sqrt [3]{4 x^3+6 x^2+\sqrt {-1+\left (4 x^3+6 x^2-1+12 c_1\right ){}^2}-1+12 c_1}}-1\right ) \\
y(x)\to \frac {1}{8} \left (2 i \left (\sqrt {3}+i\right ) \sqrt [3]{4 x^3+6 x^2+\sqrt {-1+\left (4 x^3+6 x^2-1+12 c_1\right ){}^2}-1+12 c_1}+\frac {-2-2 i \sqrt {3}}{\sqrt [3]{4 x^3+6 x^2+\sqrt {-1+\left (4 x^3+6 x^2-1+12 c_1\right ){}^2}-1+12 c_1}}-4\right ) \\
y(x)\to \frac {1}{8} \left (-2 \left (1+i \sqrt {3}\right ) \sqrt [3]{4 x^3+6 x^2+\sqrt {-1+\left (4 x^3+6 x^2-1+12 c_1\right ){}^2}-1+12 c_1}+\frac {2 i \left (\sqrt {3}+i\right )}{\sqrt [3]{4 x^3+6 x^2+\sqrt {-1+\left (4 x^3+6 x^2-1+12 c_1\right ){}^2}-1+12 c_1}}-4\right ) \\
\end{align*}