83.22.12 problem 12
Internal
problem
ID
[19266]
Book
:
A
Text
book
for
differentional
equations
for
postgraduate
students
by
Ray
and
Chaturvedi.
First
edition,
1958.
BHASKAR
press.
INDIA
Section
:
Chapter
IV.
Equations
of
the
first
order
but
not
of
the
first
degree.
Exercise
IV
(E)
at
page
63
Problem
number
:
12
Date
solved
:
Tuesday, January 28, 2025 at 01:17:52 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} a^{2} y {y^{\prime }}^{2}-4 x y^{\prime }+y&=0 \end{align*}
✓ Solution by Maple
Time used: 0.115 (sec). Leaf size: 114
dsolve(a^2*y(x)*diff(y(x),x)^2-4*x*diff(y(x),x)+y(x)=0,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= 0 \\
y \left (x \right ) &= \operatorname {RootOf}\left (-\ln \left (x \right )-\int _{}^{\textit {\_Z}}\frac {a^{2} \textit {\_a}^{2}+\sqrt {-a^{2} \textit {\_a}^{2}+4}-2}{\textit {\_a} \left (a^{2} \textit {\_a}^{2}-3\right )}d \textit {\_a} +c_{1} \right ) x \\
y \left (x \right ) &= \operatorname {RootOf}\left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}-\frac {a^{2} \textit {\_a}^{2}-\sqrt {-a^{2} \textit {\_a}^{2}+4}-2}{\textit {\_a} \left (a^{2} \textit {\_a}^{2}-3\right )}d \textit {\_a} +c_{1} \right ) x \\
\end{align*}
✓ Solution by Mathematica
Time used: 60.886 (sec). Leaf size: 1287
DSolve[a^2*y[x]*D[y[x],x]^2-4*x*D[y[x],x]+y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to -\sqrt {\frac {x^2}{a^2}+\frac {\sqrt [3]{2} a^2 x \left (x^3+2 e^{3 a^2 c_1}\right )}{\sqrt [3]{\sqrt {a^{24} e^{3 a^2 c_1} \left (-4 x^3+e^{3 a^2 c_1}\right ){}^3}-a^{12} \left (10 x^3 e^{3 a^2 c_1}+e^{6 a^2 c_1}-2 x^6\right )}}+\frac {\sqrt [3]{\sqrt {a^{24} e^{3 a^2 c_1} \left (-4 x^3+e^{3 a^2 c_1}\right ){}^3}-a^{12} \left (10 x^3 e^{3 a^2 c_1}+e^{6 a^2 c_1}-2 x^6\right )}}{\sqrt [3]{2} a^6}} \\
y(x)\to \sqrt {\frac {x^2}{a^2}+\frac {\sqrt [3]{2} a^2 x \left (x^3+2 e^{3 a^2 c_1}\right )}{\sqrt [3]{\sqrt {a^{24} e^{3 a^2 c_1} \left (-4 x^3+e^{3 a^2 c_1}\right ){}^3}-a^{12} \left (10 x^3 e^{3 a^2 c_1}+e^{6 a^2 c_1}-2 x^6\right )}}+\frac {\sqrt [3]{\sqrt {a^{24} e^{3 a^2 c_1} \left (-4 x^3+e^{3 a^2 c_1}\right ){}^3}-a^{12} \left (10 x^3 e^{3 a^2 c_1}+e^{6 a^2 c_1}-2 x^6\right )}}{\sqrt [3]{2} a^6}} \\
y(x)\to -\sqrt {\frac {x^2}{a^2}+\frac {i \left (\sqrt {3}+i\right ) a^2 x \left (x^3+2 e^{3 a^2 c_1}\right )}{2^{2/3} \sqrt [3]{\sqrt {a^{24} e^{3 a^2 c_1} \left (-4 x^3+e^{3 a^2 c_1}\right ){}^3}-a^{12} \left (10 x^3 e^{3 a^2 c_1}+e^{6 a^2 c_1}-2 x^6\right )}}-\frac {i \left (\sqrt {3}-i\right ) \sqrt [3]{\sqrt {a^{24} e^{3 a^2 c_1} \left (-4 x^3+e^{3 a^2 c_1}\right ){}^3}-a^{12} \left (10 x^3 e^{3 a^2 c_1}+e^{6 a^2 c_1}-2 x^6\right )}}{2 \sqrt [3]{2} a^6}} \\
y(x)\to \sqrt {\frac {x^2}{a^2}+\frac {i \left (\sqrt {3}+i\right ) a^2 x \left (x^3+2 e^{3 a^2 c_1}\right )}{2^{2/3} \sqrt [3]{\sqrt {a^{24} e^{3 a^2 c_1} \left (-4 x^3+e^{3 a^2 c_1}\right ){}^3}-a^{12} \left (10 x^3 e^{3 a^2 c_1}+e^{6 a^2 c_1}-2 x^6\right )}}-\frac {i \left (\sqrt {3}-i\right ) \sqrt [3]{\sqrt {a^{24} e^{3 a^2 c_1} \left (-4 x^3+e^{3 a^2 c_1}\right ){}^3}-a^{12} \left (10 x^3 e^{3 a^2 c_1}+e^{6 a^2 c_1}-2 x^6\right )}}{2 \sqrt [3]{2} a^6}} \\
y(x)\to -\sqrt {\frac {x^2}{a^2}-\frac {i \left (\sqrt {3}-i\right ) a^2 x \left (x^3+2 e^{3 a^2 c_1}\right )}{2^{2/3} \sqrt [3]{\sqrt {a^{24} e^{3 a^2 c_1} \left (-4 x^3+e^{3 a^2 c_1}\right ){}^3}-a^{12} \left (10 x^3 e^{3 a^2 c_1}+e^{6 a^2 c_1}-2 x^6\right )}}+\frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{\sqrt {a^{24} e^{3 a^2 c_1} \left (-4 x^3+e^{3 a^2 c_1}\right ){}^3}-a^{12} \left (10 x^3 e^{3 a^2 c_1}+e^{6 a^2 c_1}-2 x^6\right )}}{2 \sqrt [3]{2} a^6}} \\
y(x)\to \sqrt {\frac {x^2}{a^2}-\frac {i \left (\sqrt {3}-i\right ) a^2 x \left (x^3+2 e^{3 a^2 c_1}\right )}{2^{2/3} \sqrt [3]{\sqrt {a^{24} e^{3 a^2 c_1} \left (-4 x^3+e^{3 a^2 c_1}\right ){}^3}-a^{12} \left (10 x^3 e^{3 a^2 c_1}+e^{6 a^2 c_1}-2 x^6\right )}}+\frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{\sqrt {a^{24} e^{3 a^2 c_1} \left (-4 x^3+e^{3 a^2 c_1}\right ){}^3}-a^{12} \left (10 x^3 e^{3 a^2 c_1}+e^{6 a^2 c_1}-2 x^6\right )}}{2 \sqrt [3]{2} a^6}} \\
\end{align*}