83.23.1 problem 1
Internal
problem
ID
[19282]
Book
:
A
Text
book
for
differentional
equations
for
postgraduate
students
by
Ray
and
Chaturvedi.
First
edition,
1958.
BHASKAR
press.
INDIA
Section
:
Chapter
V.
Singular
solutions.
Exercise
V
at
page
76
Problem
number
:
1
Date
solved
:
Tuesday, January 28, 2025 at 01:24:20 PM
CAS
classification
:
[_rational, _Bernoulli]
\begin{align*} x^{2} y^{2}-3 x y y^{\prime }&=2 y^{2}+x^{3} \end{align*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 144
dsolve(x^2*y(x)^2-3*x*y(x)*diff(y(x),x)=2*y(x)^2+x^3,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= -\frac {\sqrt {\frac {{\mathrm e}^{-\frac {2 x^{2}}{3}} \left (-3 \operatorname {WhittakerM}\left (\frac {1}{12}, \frac {7}{12}, \frac {x^{2}}{3}\right ) x^{{1}/{3}} 3^{{1}/{12}} {\mathrm e}^{\frac {x^{2}}{6}}+\left (x^{{7}/{3}}+{\mathrm e}^{\frac {x^{2}}{3}} c_{1} \right ) \left (x^{2}\right )^{{1}/{12}}\right ) x^{{4}/{3}}}{\left (x^{2}\right )^{{1}/{12}}}}\, {\mathrm e}^{\frac {x^{2}}{3}}}{x^{{4}/{3}}} \\
y \left (x \right ) &= \frac {\sqrt {\frac {{\mathrm e}^{-\frac {2 x^{2}}{3}} \left (-3 \operatorname {WhittakerM}\left (\frac {1}{12}, \frac {7}{12}, \frac {x^{2}}{3}\right ) x^{{1}/{3}} 3^{{1}/{12}} {\mathrm e}^{\frac {x^{2}}{6}}+\left (x^{{7}/{3}}+{\mathrm e}^{\frac {x^{2}}{3}} c_{1} \right ) \left (x^{2}\right )^{{1}/{12}}\right ) x^{{4}/{3}}}{\left (x^{2}\right )^{{1}/{12}}}}\, {\mathrm e}^{\frac {x^{2}}{3}}}{x^{{4}/{3}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 7.582 (sec). Leaf size: 113
DSolve[x^2*y[x]^2-3*x*y[x]*D[y[x],x]==2*y[x]^2+x^3,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to -\sqrt {\frac {e^{\frac {x^2}{3}} \left (3 \sqrt [6]{3} \left (x^2\right )^{5/6} \Gamma \left (\frac {13}{6},\frac {x^2}{3}\right )+c_1 x^{5/3}\right )}{x^3}} \\
y(x)\to \sqrt {\frac {e^{\frac {x^2}{3}} \left (3 \sqrt [6]{3} \left (x^2\right )^{5/6} \Gamma \left (\frac {13}{6},\frac {x^2}{3}\right )+c_1 x^{5/3}\right )}{x^3}} \\
\end{align*}