34.3.6 problem 23 (j)
Internal
problem
ID
[7897]
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)
Date
solved
:
Tuesday, September 30, 2025 at 05:09:35 PM
CAS
classification
:
[[_homogeneous, `class A`], _exact, _rational, _dAlembert]
\begin{align*} 2 u^{2}+2 u v+\left (u^{2}+v^{2}\right ) v^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.018 (sec). Leaf size: 317
ode:=2*u^2+2*u*v(u)+(u^2+v(u)^2)*diff(v(u),u) = 0;
dsolve(ode,v(u), singsol=all);
\begin{align*}
v &= -\frac {2 \left (u^{2} c_1 -\frac {\left (4-8 u^{3} c_1^{{3}/{2}}+4 \sqrt {8 u^{6} c_1^{3}-4 u^{3} c_1^{{3}/{2}}+1}\right )^{{2}/{3}}}{4}\right )}{\left (4-8 u^{3} c_1^{{3}/{2}}+4 \sqrt {8 u^{6} c_1^{3}-4 u^{3} c_1^{{3}/{2}}+1}\right )^{{1}/{3}} \sqrt {c_1}} \\
v &= -\frac {\left (1+i \sqrt {3}\right ) \left (4-8 u^{3} c_1^{{3}/{2}}+4 \sqrt {8 u^{6} c_1^{3}-4 u^{3} c_1^{{3}/{2}}+1}\right )^{{1}/{3}}}{4 \sqrt {c_1}}-\frac {\sqrt {c_1}\, \left (i \sqrt {3}-1\right ) u^{2}}{\left (4-8 u^{3} c_1^{{3}/{2}}+4 \sqrt {8 u^{6} c_1^{3}-4 u^{3} c_1^{{3}/{2}}+1}\right )^{{1}/{3}}} \\
v &= \frac {4 i \sqrt {3}\, c_1 \,u^{2}+i \sqrt {3}\, \left (4-8 u^{3} c_1^{{3}/{2}}+4 \sqrt {8 u^{6} c_1^{3}-4 u^{3} c_1^{{3}/{2}}+1}\right )^{{2}/{3}}+4 u^{2} c_1 -\left (4-8 u^{3} c_1^{{3}/{2}}+4 \sqrt {8 u^{6} c_1^{3}-4 u^{3} c_1^{{3}/{2}}+1}\right )^{{2}/{3}}}{4 \left (4-8 u^{3} c_1^{{3}/{2}}+4 \sqrt {8 u^{6} c_1^{3}-4 u^{3} c_1^{{3}/{2}}+1}\right )^{{1}/{3}} \sqrt {c_1}} \\
\end{align*}
✓ Mathematica. Time used: 25.358 (sec). Leaf size: 622
ode=2*(u^2+u*v[u])+(u^2+v[u]^2)*D[ v[u],u]==0;
ic={};
DSolve[{ode,ic},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 {\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 \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 \frac {\left (\sqrt {2} \sqrt {u^6}-u^3\right )^{2/3}-u^2}{\sqrt [3]{\sqrt {2} \sqrt {u^6}-u^3}}\\ v(u)&\to \frac {i \sqrt {3} u^2+u^2+i \sqrt {3} \left (\sqrt {2} \sqrt {u^6}-u^3\right )^{2/3}-\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 {2 \sqrt [3]{2} \left (\sqrt {2} \sqrt {u^6}-u^3\right )^{2/3} \text {Root}\left [\text {$\#$1}^3-32\&,2\right ]+2^{2/3} u^2 \text {Root}\left [\text {$\#$1}^3+128\&,2\right ]}{8 \sqrt [3]{\sqrt {2} \sqrt {u^6}-u^3}} \end{align*}
✗ Sympy
from sympy import *
u = symbols("u")
v = Function("v")
ode = Eq(2*u**2 + 2*u*v(u) + (u**2 + v(u)**2)*Derivative(v(u), u),0)
ics = {}
dsolve(ode,func=v(u),ics=ics)
Timed Out