53.2.7 problem 14
Internal
problem
ID
[8460]
Book
:
Elementary
differential
equations.
By
Earl
D.
Rainville,
Phillip
E.
Bedient.
Macmilliam
Publishing
Co.
NY.
6th
edition.
1981.
Section
:
CHAPTER
16.
Nonlinear
equations.
Section
97.
The
p-discriminant
equation.
EXERCISES
Page
314
Problem
number
:
14
Date
solved
:
Monday, January 27, 2025 at 04:02:18 PM
CAS
classification
:
[[_1st_order, _with_linear_symmetries], _rational]
\begin{align*} 4 y^{3} {y^{\prime }}^{2}+4 x y^{\prime }+y&=0 \end{align*}
✓ Solution by Maple
Time used: 0.187 (sec). Leaf size: 307
dsolve(4*y(x)^3*diff(y(x),x)^2+4*x*diff(y(x),x)+y(x)=0,y(x), singsol=all)
\begin{align*}
y &= 0 \\
\frac {\left (\int _{\textit {\_b}}^{x}\frac {-2 \textit {\_a} +\sqrt {-y^{4}+\textit {\_a}^{2}}}{y^{4}+3 \textit {\_a}^{2}}d \textit {\_a} \right )}{2}-\int _{}^{y}\frac {\left (1+\left (\textit {\_f}^{4}-\sqrt {-\textit {\_f}^{4}+x^{2}}\, x +x^{2}\right ) \left (\int _{\textit {\_b}}^{x}\frac {\textit {\_f}^{4}+4 \sqrt {-\textit {\_f}^{4}+\textit {\_a}^{2}}\, \textit {\_a} -5 \textit {\_a}^{2}}{\sqrt {-\textit {\_f}^{4}+\textit {\_a}^{2}}\, \left (\textit {\_f}^{4}+3 \textit {\_a}^{2}\right )^{2}}d \textit {\_a} \right )\right ) \textit {\_f}^{3}}{\textit {\_f}^{4}-\sqrt {-\textit {\_f}^{4}+x^{2}}\, x +x^{2}}d \textit {\_f} +c_{1} &= 0 \\
-\frac {\left (\int _{\textit {\_b}}^{x}\frac {2 \textit {\_a} +\sqrt {-y^{4}+\textit {\_a}^{2}}}{y^{4}+3 \textit {\_a}^{2}}d \textit {\_a} \right )}{2}-\int _{}^{y}\frac {\textit {\_f}^{3} \left (1+\left (\textit {\_f}^{4}+\sqrt {-\textit {\_f}^{4}+x^{2}}\, x +x^{2}\right ) \left (\int _{\textit {\_b}}^{x}\frac {-\textit {\_f}^{4}+5 \textit {\_a}^{2}+4 \sqrt {-\textit {\_f}^{4}+\textit {\_a}^{2}}\, \textit {\_a}}{\sqrt {-\textit {\_f}^{4}+\textit {\_a}^{2}}\, \left (\textit {\_f}^{4}+3 \textit {\_a}^{2}\right )^{2}}d \textit {\_a} \right )\right )}{\textit {\_f}^{4}+\sqrt {-\textit {\_f}^{4}+x^{2}}\, x +x^{2}}d \textit {\_f} +c_{1} &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 60.299 (sec). Leaf size: 2815
DSolve[4*y[x]^3*(D[y[x],x])^2+4*x*D[y[x],x]+y[x]==0,y[x],x,IncludeSingularSolutions -> True]
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