77.1.90 problem 117 (page 177)
Internal
problem
ID
[17980]
Book
:
V.V.
Stepanov,
A
course
of
differential
equations
(in
Russian),
GIFML.
Moscow
(1958)
Section
:
All
content
Problem
number
:
117
(page
177)
Date
solved
:
Tuesday, January 28, 2025 at 08:28:23 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} y^{2} \left (x^{2} y^{\prime \prime }-x y^{\prime }+y\right )&=x^{3} \end{align*}
✓ Solution by Maple
Time used: 0.329 (sec). Leaf size: 685
dsolve(y(x)^2*( x^2*diff(y(x),x$2)-x*diff(y(x),x) + y(x) )=x^3,y(x), singsol=all)
\begin{align*}
y &= \frac {x c_{1} \left ({\mathrm e}^{2 \operatorname {RootOf}\left (-{\mathrm e}^{2 \textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{2}-2 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} c_{1}^{3}+\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4}+2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) \ln \left (x \right )-2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{2} \right )}+2 c_{1} {\mathrm e}^{\operatorname {RootOf}\left (-{\mathrm e}^{2 \textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{2}-2 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} c_{1}^{3}+\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4}+2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) \ln \left (x \right )-2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{2} \right )}+c_{1}^{2}\right ) {\mathrm e}^{-\operatorname {RootOf}\left (-{\mathrm e}^{2 \textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{2}-2 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} c_{1}^{3}+\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4}+2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) \ln \left (x \right )-2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{2} \right )}}{2} \\
y &= \frac {x c_{1} \left ({\mathrm e}^{2 \operatorname {RootOf}\left (-{\mathrm e}^{2 \textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{2}+2 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} c_{1}^{3}+\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4}-2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) \ln \left (x \right )+2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{2} \right )}+2 c_{1} {\mathrm e}^{\operatorname {RootOf}\left (-{\mathrm e}^{2 \textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{2}+2 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} c_{1}^{3}+\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4}-2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) \ln \left (x \right )+2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{2} \right )}+c_{1}^{2}\right ) {\mathrm e}^{-\operatorname {RootOf}\left (-{\mathrm e}^{2 \textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{2}+2 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} c_{1}^{3}+\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4}-2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) \ln \left (x \right )+2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{2} \right )}}{2} \\
y &= \frac {x c_{1} \left ({\mathrm e}^{2 \operatorname {RootOf}\left (-{\mathrm e}^{2 \textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{2}-2 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} c_{1}^{3}+\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4}-2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) \ln \left (x \right )+2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{2} \right )}+2 c_{1} {\mathrm e}^{\operatorname {RootOf}\left (-{\mathrm e}^{2 \textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{2}-2 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} c_{1}^{3}+\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4}-2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) \ln \left (x \right )+2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{2} \right )}+c_{1}^{2}\right ) {\mathrm e}^{-\operatorname {RootOf}\left (-{\mathrm e}^{2 \textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{2}-2 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} c_{1}^{3}+\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4}-2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) \ln \left (x \right )+2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{2} \right )}}{2} \\
y &= \frac {x c_{1} \left ({\mathrm e}^{2 \operatorname {RootOf}\left (-{\mathrm e}^{2 \textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{2}+2 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} c_{1}^{3}+\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4}+2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) \ln \left (x \right )-2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{2} \right )}+2 c_{1} {\mathrm e}^{\operatorname {RootOf}\left (-{\mathrm e}^{2 \textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{2}+2 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} c_{1}^{3}+\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4}+2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) \ln \left (x \right )-2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{2} \right )}+c_{1}^{2}\right ) {\mathrm e}^{-\operatorname {RootOf}\left (-{\mathrm e}^{2 \textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{2}+2 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} c_{1}^{3}+\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4}+2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) \ln \left (x \right )-2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{2} \right )}}{2} \\
\end{align*}
✓ Solution by Mathematica
Time used: 2.634 (sec). Leaf size: 268
DSolve[y[x]^2*( x^2*D[y[x],{x,2}]-x*D[y[x],x] + y[x] )==x^3,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
\text {Solve}\left [\log (x)-\frac {2 x^{3/2} \sqrt {2-\frac {c_1 y(x)}{x}} \arcsin \left (\frac {\sqrt {c_1} \sqrt {y(x)}}{\sqrt {2} \sqrt {x}}\right )-2 \sqrt {c_1} x \sqrt {y(x)}+c_1{}^{3/2} y(x)^{3/2}}{c_1{}^{3/2} x \sqrt {-\frac {x^2}{y(x)^2}} \sqrt {y(x)} \sqrt {\frac {y(x) (2 x-c_1 y(x))}{x^2}}}&=c_2,y(x)\right ] \\
\text {Solve}\left [\log (x)+\frac {2 x^{3/2} \sqrt {2-\frac {c_1 y(x)}{x}} \arcsin \left (\frac {\sqrt {c_1} \sqrt {y(x)}}{\sqrt {2} \sqrt {x}}\right )-2 \sqrt {c_1} x \sqrt {y(x)}+c_1{}^{3/2} y(x)^{3/2}}{c_1{}^{3/2} x \sqrt {-\frac {x^2}{y(x)^2}} \sqrt {y(x)} \sqrt {\frac {y(x) (2 x-c_1 y(x))}{x^2}}}&=c_2,y(x)\right ] \\
\end{align*}