12.7.23 problem 24
Internal
problem
ID
[1733]
Book
:
Elementary
differential
equations
with
boundary
value
problems.
William
F.
Trench.
Brooks/Cole
2001
Section
:
Chapter
2,
First
order
equations.
Exact
equations.
Integrating
factors.
Section
2.6
Page
91
Problem
number
:
24
Date
solved
:
Monday, January 27, 2025 at 05:33:44 AM
CAS
classification
:
[[_homogeneous, `class G`], _rational]
\begin{align*} x^{4} y^{3}+y+\left (x^{5} y^{2}-x \right ) y^{\prime }&=0 \end{align*}
✓ Solution by Maple
Time used: 0.033 (sec). Leaf size: 238
dsolve((x^4*y(x)^3+y(x))+(x^5*y(x)^2-x)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*}
y &= \frac {\left (4 x^{3}+4 \sqrt {x^{6}-4 c_1^{3}}\right )^{{2}/{3}}+4 c_1}{2 x^{2} \left (4 x^{3}+4 \sqrt {x^{6}-4 c_1^{3}}\right )^{{1}/{3}} \sqrt {c_1}} \\
y &= \frac {-i \sqrt {3}\, \left (4 x^{3}+4 \sqrt {x^{6}-4 c_1^{3}}\right )^{{2}/{3}}+4 i \sqrt {3}\, c_1 -\left (4 x^{3}+4 \sqrt {x^{6}-4 c_1^{3}}\right )^{{2}/{3}}-4 c_1}{4 x^{2} \left (4 x^{3}+4 \sqrt {x^{6}-4 c_1^{3}}\right )^{{1}/{3}} \sqrt {c_1}} \\
y &= -\frac {-i \sqrt {3}\, \left (4 x^{3}+4 \sqrt {x^{6}-4 c_1^{3}}\right )^{{2}/{3}}+4 i \sqrt {3}\, c_1 +\left (4 x^{3}+4 \sqrt {x^{6}-4 c_1^{3}}\right )^{{2}/{3}}+4 c_1}{4 x^{2} \left (4 x^{3}+4 \sqrt {x^{6}-4 c_1^{3}}\right )^{{1}/{3}} \sqrt {c_1}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 44.816 (sec). Leaf size: 300
DSolve[(x^4*y[x]^3+y[x])+(x^5*y[x]^2-x)*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {2+\frac {\sqrt [3]{2} \left (3 c_1 x^9+\sqrt {x^{12} \left (-4+9 c_1{}^2 x^6\right )}\right ){}^{2/3}}{x^4}}{2^{2/3} \sqrt [3]{3 c_1 x^9+\sqrt {x^{12} \left (-4+9 c_1{}^2 x^6\right )}}} \\
y(x)\to \frac {i \left (\left (\sqrt {3}+i\right ) \left (6 c_1 x^9+2 \sqrt {x^{12} \left (-4+9 c_1{}^2 x^6\right )}\right ){}^{2/3}-2 \sqrt [3]{2} \left (\sqrt {3}-i\right ) x^4\right )}{4 x^4 \sqrt [3]{3 c_1 x^9+\sqrt {x^{12} \left (-4+9 c_1{}^2 x^6\right )}}} \\
y(x)\to \frac {2 i \sqrt [3]{2} \left (\sqrt {3}+i\right ) x^4+\left (-1-i \sqrt {3}\right ) \left (6 c_1 x^9+2 \sqrt {x^{12} \left (-4+9 c_1{}^2 x^6\right )}\right ){}^{2/3}}{4 x^4 \sqrt [3]{3 c_1 x^9+\sqrt {x^{12} \left (-4+9 c_1{}^2 x^6\right )}}} \\
\end{align*}