8.5.35 problem 35

Internal problem ID [763]
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
Date solved : Monday, January 27, 2025 at 03:04:06 AM
CAS classification : [_exact]

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 305

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

\begin{align*} y &= \frac {\left (-3 x^{4}-12 c_1 +\sqrt {64 \ln \left (x \right )^{3}+9 \left (x^{4}+4 c_1 \right )^{2}}\right )^{{2}/{3}}-4 \ln \left (x \right )}{2 \left (-3 x^{4}-12 c_1 +\sqrt {64 \ln \left (x \right )^{3}+9 \left (x^{4}+4 c_1 \right )^{2}}\right )^{{1}/{3}}} \\ y &= \frac {i \left (-\left (-3 x^{4}-12 c_1 +\sqrt {64 \ln \left (x \right )^{3}+9 \left (x^{4}+4 c_1 \right )^{2}}\right )^{{2}/{3}}-4 \ln \left (x \right )\right ) \sqrt {3}-\left (-3 x^{4}-12 c_1 +\sqrt {64 \ln \left (x \right )^{3}+9 \left (x^{4}+4 c_1 \right )^{2}}\right )^{{2}/{3}}+4 \ln \left (x \right )}{4 \left (-3 x^{4}-12 c_1 +\sqrt {64 \ln \left (x \right )^{3}+9 \left (x^{4}+4 c_1 \right )^{2}}\right )^{{1}/{3}}} \\ y &= \frac {i \left (\left (-3 x^{4}-12 c_1 +\sqrt {64 \ln \left (x \right )^{3}+9 \left (x^{4}+4 c_1 \right )^{2}}\right )^{{2}/{3}}+4 \ln \left (x \right )\right ) \sqrt {3}-\left (-3 x^{4}-12 c_1 +\sqrt {64 \ln \left (x \right )^{3}+9 \left (x^{4}+4 c_1 \right )^{2}}\right )^{{2}/{3}}+4 \ln \left (x \right )}{4 \left (-3 x^{4}-12 c_1 +\sqrt {64 \ln \left (x \right )^{3}+9 \left (x^{4}+4 c_1 \right )^{2}}\right )^{{1}/{3}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.815 (sec). Leaf size: 307

DSolve[x^3+y[x]/x+(Log[x]+y[x]^2)*D[y[x],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*}