40.5.6 problem 22
Internal
problem
ID
[6671]
Book
:
Schaums
Outline.
Theory
and
problems
of
Differential
Equations,
1st
edition.
Frank
Ayres.
McGraw
Hill
1952
Section
:
Chapter
9.
Equations
of
first
order
and
higher
degree.
Supplemetary
problems.
Page
65
Problem
number
:
22
Date
solved
:
Monday, January 27, 2025 at 02:18:49 PM
CAS
classification
:
[[_1st_order, _with_linear_symmetries], _rational]
\begin{align*} y^{2} {y^{\prime }}^{2}+3 x y^{\prime }-y&=0 \end{align*}
✓ Solution by Maple
Time used: 0.400 (sec). Leaf size: 124
dsolve(y(x)^2*diff(y(x),x)^2+3*x*diff(y(x),x)-y(x)=0,y(x), singsol=all)
\begin{align*}
y &= \frac {18^{{1}/{3}} \left (-x^{2}\right )^{{1}/{3}}}{2} \\
y &= -\frac {2^{{1}/{3}} \left (-x^{2}\right )^{{1}/{3}} \left (3 i 3^{{1}/{6}}+3^{{2}/{3}}\right )}{4} \\
y &= \frac {2^{{1}/{3}} \left (-x^{2}\right )^{{1}/{3}} \left (-3^{{2}/{3}}+3 i 3^{{1}/{6}}\right )}{4} \\
y &= 0 \\
y &= \operatorname {RootOf}\left (-2 \ln \left (x \right )+3 \left (\int _{}^{\textit {\_Z}}-\frac {4 \textit {\_a}^{3}-3 \sqrt {4 \textit {\_a}^{3}+9}+9}{\textit {\_a} \left (4 \textit {\_a}^{3}+9\right )}d \textit {\_a} \right )+2 c_1 \right ) x^{{2}/{3}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.540 (sec). Leaf size: 204
DSolve[y[x]^2*D[y[x],x]^2+3*x*D[y[x],x]-y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
\text {Solve}\left [\frac {3}{2} \log (y(x))-\frac {2 \sqrt {\frac {9 x^2}{4 y(x)^3}+1} \text {arcsinh}\left (\frac {3}{2} x \sqrt {\frac {1}{y(x)^3}}\right )}{\sqrt {\frac {1}{y(x)^3}} \sqrt {9 x^2+4 y(x)^3}}&=c_1,y(x)\right ] \\
\text {Solve}\left [\frac {2 \sqrt {\frac {9 x^2}{4 y(x)^3}+1} \text {arcsinh}\left (\frac {3}{2} x \sqrt {\frac {1}{y(x)^3}}\right )}{\sqrt {\frac {1}{y(x)^3}} \sqrt {9 x^2+4 y(x)^3}}+\frac {3}{2} \log (y(x))&=c_1,y(x)\right ] \\
y(x)\to -\left (-\frac {3}{2}\right )^{2/3} x^{2/3} \\
y(x)\to -\left (\frac {3}{2}\right )^{2/3} x^{2/3} \\
y(x)\to \frac {\sqrt [3]{-1} 3^{2/3} x^{2/3}}{2^{2/3}} \\
\end{align*}