83.46.11 problem Ex 11 page 75
Internal
problem
ID
[19576]
Book
:
A
Text
book
for
differentional
equations
for
postgraduate
students
by
Ray
and
Chaturvedi.
First
edition,
1958.
BHASKAR
press.
INDIA
Section
:
Book
Solved
Excercises.
Chapter
V.
Singular
solutions
Problem
number
:
Ex
11
page
75
Date
solved
:
Tuesday, January 28, 2025 at 01:57:36 PM
CAS
classification
:
[[_homogeneous, `class A`], _dAlembert]
\begin{align*} {y^{\prime }}^{2} y^{2} \cos \left (a \right )^{2}-2 y^{\prime } x y \sin \left (a \right )^{2}+y^{2}-x^{2} \sin \left (a \right )^{2}&=0 \end{align*}
✓ Solution by Maple
Time used: 0.117 (sec). Leaf size: 167
dsolve(diff(y(x),x)^2*y(x)^2*cos(a)^2-2*diff(y(x),x)*x*y(x)*sin(a)^2+y(x)^2-x^2*sin(a)^2=0,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= \operatorname {RootOf}\left (-\ln \left (x \right )-\int _{}^{\textit {\_Z}}\frac {\left (\textit {\_a}^{2} \cos \left (a \right )^{2}-\sin \left (a \right )^{2}-\sqrt {-\textit {\_a}^{2} \cos \left (a \right )^{2}+\sin \left (a \right )^{2}}\right ) \textit {\_a}}{\cos \left (a \right )^{2} \textit {\_a}^{4}+2 \textit {\_a}^{2} \cos \left (a \right )^{2}-\sin \left (a \right )^{2}-\textit {\_a}^{2}}d \textit {\_a} +c_{1} \right ) x \\
y \left (x \right ) &= \operatorname {RootOf}\left (-\ln \left (x \right )-\int _{}^{\textit {\_Z}}\frac {\left (\textit {\_a}^{2} \cos \left (a \right )^{2}-\sin \left (a \right )^{2}+\sqrt {-\textit {\_a}^{2} \cos \left (a \right )^{2}+\sin \left (a \right )^{2}}\right ) \textit {\_a}}{\cos \left (a \right )^{2} \textit {\_a}^{4}+2 \textit {\_a}^{2} \cos \left (a \right )^{2}-\sin \left (a \right )^{2}-\textit {\_a}^{2}}d \textit {\_a} +c_{1} \right ) x \\
\end{align*}
✓ Solution by Mathematica
Time used: 38.867 (sec). Leaf size: 281
DSolve[D[y[x],x]^2*y[x]^2*Cos[a]^2-2*D[y[x],x]*x*y[x]*Sin[a]^2+y[x]^2-x^2*Sin[a]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to -\frac {\sqrt {x^2 (-\cos (2 a))+2 \sqrt {2} x e^{2 c_1 \cos ^2(a)}-e^{4 c_1 \cos ^2(a)}-x^2}}{\sqrt {\cos (2 a)+1}} \\
y(x)\to \frac {\sqrt {x^2 (-\cos (2 a))+2 \sqrt {2} x e^{2 c_1 \cos ^2(a)}-e^{4 c_1 \cos ^2(a)}-x^2}}{\sqrt {\cos (2 a)+1}} \\
y(x)\to -\frac {(-1)^{3/4} \sqrt {-i x^2 \cos (2 a)+2 \sqrt {2} x e^{2 c_1 \cos ^2(a)}+i e^{4 c_1 \cos ^2(a)}-i x^2}}{\sqrt {\cos (2 a)+1}} \\
y(x)\to \frac {(-1)^{3/4} \sqrt {-i x^2 \cos (2 a)+2 \sqrt {2} x e^{2 c_1 \cos ^2(a)}+i e^{4 c_1 \cos ^2(a)}-i x^2}}{\sqrt {\cos (2 a)+1}} \\
\end{align*}