Internal problem ID [5267]
Book: Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres.
McGraw Hill 1952
Section: Chapter 5. Equations of first order and first degree (Exact equations). Supplemetary
problems. Page 33
Problem number: 23 (j).
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, `class A`], _exact, _rational, _dAlembert]
\[ \boxed {2 u v+\left (u^{2}+v^{2}\right ) v^{\prime }=-2 u^{2}} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 417
dsolve(2*(u^2+u*v(u))+(u^2+v(u)^2)*diff(v(u),u)=0,v(u), singsol=all)
\begin{align*} v \left (u \right ) = \frac {\frac {\left (4-8 u^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {8 u^{6} c_{1}^{3}-4 u^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}{2}-\frac {2 u^{2} c_{1}}{\left (4-8 u^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {8 u^{6} c_{1}^{3}-4 u^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}}{\sqrt {c_{1}}} v \left (u \right ) = \frac {-\frac {\left (4-8 u^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {8 u^{6} c_{1}^{3}-4 u^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}{4}+\frac {u^{2} c_{1}}{\left (4-8 u^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {8 u^{6} c_{1}^{3}-4 u^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (4-8 u^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {8 u^{6} c_{1}^{3}-4 u^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}{2}+\frac {2 u^{2} c_{1}}{\left (4-8 u^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {8 u^{6} c_{1}^{3}-4 u^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}\right )}{2}}{\sqrt {c_{1}}} v \left (u \right ) = \frac {-\frac {\left (4-8 u^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {8 u^{6} c_{1}^{3}-4 u^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}{4}+\frac {u^{2} c_{1}}{\left (4-8 u^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {8 u^{6} c_{1}^{3}-4 u^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (4-8 u^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {8 u^{6} c_{1}^{3}-4 u^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}{2}+\frac {2 u^{2} c_{1}}{\left (4-8 u^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {8 u^{6} c_{1}^{3}-4 u^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}\right )}{2}}{\sqrt {c_{1}}} \end{align*}
✓ Solution by Mathematica
Time used: 15.565 (sec). Leaf size: 593
DSolve[2*(u^2+u*v[u])+(u^2+v[u]^2)*v'[u]==0,v[u],u,IncludeSingularSolutions -> True]
\begin{align*} v(u)\to \frac {\sqrt [3]{-2 u^3+\sqrt {8 u^6-4 e^{3 c_1} u^3+e^{6 c_1}}+e^{3 c_1}}}{\sqrt [3]{2}}-\frac {\sqrt [3]{2} u^2}{\sqrt [3]{-2 u^3+\sqrt {8 u^6-4 e^{3 c_1} u^3+e^{6 c_1}}+e^{3 c_1}}} v(u)\to \frac {\sqrt [3]{2} \left (2+2 i \sqrt {3}\right ) u^2+i 2^{2/3} \left (\sqrt {3}+i\right ) \left (-2 u^3+\sqrt {8 u^6-4 e^{3 c_1} u^3+e^{6 c_1}}+e^{3 c_1}\right ){}^{2/3}}{4 \sqrt [3]{-2 u^3+\sqrt {8 u^6-4 e^{3 c_1} u^3+e^{6 c_1}}+e^{3 c_1}}} v(u)\to \frac {\left (1-i \sqrt {3}\right ) u^2}{2^{2/3} \sqrt [3]{-2 u^3+\sqrt {8 u^6-4 e^{3 c_1} u^3+e^{6 c_1}}+e^{3 c_1}}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-2 u^3+\sqrt {8 u^6-4 e^{3 c_1} u^3+e^{6 c_1}}+e^{3 c_1}}}{2 \sqrt [3]{2}} v(u)\to \sqrt [3]{\sqrt {2} \sqrt {u^6}-u^3}-\frac {u^2}{\sqrt [3]{\sqrt {2} \sqrt {u^6}-u^3}} v(u)\to \frac {\left (1-i \sqrt {3}\right ) u^2+\left (-1-i \sqrt {3}\right ) \left (\sqrt {2} \sqrt {u^6}-u^3\right )^{2/3}}{2 \sqrt [3]{\sqrt {2} \sqrt {u^6}-u^3}} v(u)\to \frac {\left (1+i \sqrt {3}\right ) u^2+i \left (\sqrt {3}+i\right ) \left (\sqrt {2} \sqrt {u^6}-u^3\right )^{2/3}}{2 \sqrt [3]{\sqrt {2} \sqrt {u^6}-u^3}} \end{align*}