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
:
Saturday, March 29, 2025 at 10:20:26 PM
CAS
classification
:
[_exact]
\begin{align*} x^{3}+\frac {y}{x}+\left (\ln \left (x \right )+y^{2}\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.007 (sec). Leaf size: 305
ode:=x^3+y(x)/x+(ln(x)+y(x)^2)*diff(y(x),x) = 0;
dsolve(ode,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*}
✓ Mathematica. Time used: 1.815 (sec). Leaf size: 307
ode=x^3+y[x]/x+(Log[x]+y[x]^2)*D[y[x],x] == 0;
ic={};
DSolve[{ode,ic},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*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x**3 + (y(x)**2 + log(x))*Derivative(y(x), x) + y(x)/x,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE Derivative(y(x), x) - (-x**4 - y(x))/(x*(y(x)**2 + log(x))) cannot be solved by the factorable group method