83.46.11 problem Ex 11 page 75

Internal problem ID [19505]
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 : Monday, March 31, 2025 at 07:26:36 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*}

✓ Maple. Time used: 0.112 (sec). Leaf size: 167

ode:=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; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \operatorname {RootOf}\left (-\ln \left (x \right )-\int _{}^{\textit {\_Z}}\frac {\textit {\_a} \left (\cos \left (a \right )^{2} \textit {\_a}^{2}-\sin \left (a \right )^{2}-\sqrt {-\cos \left (a \right )^{2} \textit {\_a}^{2}+\sin \left (a \right )^{2}}\right )}{\cos \left (a \right )^{2} \textit {\_a}^{4}+2 \cos \left (a \right )^{2} \textit {\_a}^{2}-\sin \left (a \right )^{2}-\textit {\_a}^{2}}d \textit {\_a} +c_1 \right ) x \\ y &= \operatorname {RootOf}\left (-\ln \left (x \right )-\int _{}^{\textit {\_Z}}\frac {\textit {\_a} \left (\cos \left (a \right )^{2} \textit {\_a}^{2}-\sin \left (a \right )^{2}+\sqrt {-\cos \left (a \right )^{2} \textit {\_a}^{2}+\sin \left (a \right )^{2}}\right )}{\cos \left (a \right )^{2} \textit {\_a}^{4}+2 \cos \left (a \right )^{2} \textit {\_a}^{2}-\sin \left (a \right )^{2}-\textit {\_a}^{2}}d \textit {\_a} +c_1 \right ) x \\ \end{align*}

✓ Mathematica. Time used: 38.867 (sec). Leaf size: 281

ode=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; 
ic={}; 
DSolve[{ode,ic},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*}

✓ Sympy. Time used: 170.132 (sec). Leaf size: 495

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-x**2*sin(a)**2 - 2*x*y(x)*sin(a)**2*Derivative(y(x), x) + y(x)**2*cos(a)**2*Derivative(y(x), x)**2 + y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \text {Solution too large to show} \]