Internal problem ID [113]
Book: Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section: Section 1.6, Substitution methods and exact equations. Page 74
Problem number: 35.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_exact]
Solve \begin {gather*} \boxed {x^{3}+\frac {y}{x}+\left (\ln \relax (x )+y^{2}\right ) y^{\prime }=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 415
dsolve(x^3+y(x)/x+(ln(x)+y(x)^2)*diff(y(x),x) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\left (-3 x^{4}-12 c_{1}+\sqrt {64 \ln \relax (x )^{3}+9 x^{8}+72 x^{4} c_{1}+144 c_{1}^{2}}\right )^{\frac {1}{3}}}{2}-\frac {2 \ln \relax (x )}{\left (-3 x^{4}-12 c_{1}+\sqrt {64 \ln \relax (x )^{3}+9 x^{8}+72 x^{4} c_{1}+144 c_{1}^{2}}\right )^{\frac {1}{3}}} \\ y \relax (x ) = -\frac {\left (-3 x^{4}-12 c_{1}+\sqrt {64 \ln \relax (x )^{3}+9 x^{8}+72 x^{4} c_{1}+144 c_{1}^{2}}\right )^{\frac {1}{3}}}{4}+\frac {\ln \relax (x )}{\left (-3 x^{4}-12 c_{1}+\sqrt {64 \ln \relax (x )^{3}+9 x^{8}+72 x^{4} c_{1}+144 c_{1}^{2}}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (-3 x^{4}-12 c_{1}+\sqrt {64 \ln \relax (x )^{3}+9 x^{8}+72 x^{4} c_{1}+144 c_{1}^{2}}\right )^{\frac {1}{3}}}{2}+\frac {2 \ln \relax (x )}{\left (-3 x^{4}-12 c_{1}+\sqrt {64 \ln \relax (x )^{3}+9 x^{8}+72 x^{4} c_{1}+144 c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (x ) = -\frac {\left (-3 x^{4}-12 c_{1}+\sqrt {64 \ln \relax (x )^{3}+9 x^{8}+72 x^{4} c_{1}+144 c_{1}^{2}}\right )^{\frac {1}{3}}}{4}+\frac {\ln \relax (x )}{\left (-3 x^{4}-12 c_{1}+\sqrt {64 \ln \relax (x )^{3}+9 x^{8}+72 x^{4} c_{1}+144 c_{1}^{2}}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (-3 x^{4}-12 c_{1}+\sqrt {64 \ln \relax (x )^{3}+9 x^{8}+72 x^{4} c_{1}+144 c_{1}^{2}}\right )^{\frac {1}{3}}}{2}+\frac {2 \ln \relax (x )}{\left (-3 x^{4}-12 c_{1}+\sqrt {64 \ln \relax (x )^{3}+9 x^{8}+72 x^{4} c_{1}+144 c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.817 (sec). Leaf size: 307
DSolve[x^3+y[x]/x+(Log[x]+y[x]^2)*y'[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {-4 \log (x)+\left (-3 x^4+\sqrt {64 \log ^3(x)+9 \left (x^4-4 c_1\right ){}^2}+12 c_1\right ){}^{2/3}}{2 \sqrt [3]{-3 x^4+\sqrt {64 \log ^3(x)+9 \left (x^4-4 c_1\right ){}^2}+12 c_1}} \\ y(x)\to \frac {i \left (\sqrt {3}+i\right ) \left (-3 x^4+\sqrt {64 \log ^3(x)+9 \left (x^4-4 c_1\right ){}^2}+12 c_1\right ){}^{2/3}+\left (4+4 i \sqrt {3}\right ) \log (x)}{4 \sqrt [3]{-3 x^4+\sqrt {64 \log ^3(x)+9 \left (x^4-4 c_1\right ){}^2}+12 c_1}} \\ y(x)\to \frac {\left (-1-i \sqrt {3}\right ) \left (-3 x^4+\sqrt {64 \log ^3(x)+9 \left (x^4-4 c_1\right ){}^2}+12 c_1\right ){}^{2/3}+\left (4-4 i \sqrt {3}\right ) \log (x)}{4 \sqrt [3]{-3 x^4+\sqrt {64 \log ^3(x)+9 \left (x^4-4 c_1\right ){}^2}+12 c_1}} \\ \end{align*}