54.1.331 problem 338

Internal problem ID [11645]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 338
Date solved : Tuesday, September 30, 2025 at 09:53:13 PM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

\begin{align*} \left (y \sqrt {y^{2}+x^{2}}+\left (y^{2}-x^{2}\right ) \sin \left (\alpha \right )-2 x y \cos \left (\alpha \right )\right ) y^{\prime }+x \sqrt {y^{2}+x^{2}}+2 x y \sin \left (\alpha \right )+\left (y^{2}-x^{2}\right ) \cos \left (\alpha \right )&=0 \end{align*}

✓ Maple. Time used: 0.518 (sec). Leaf size: 128

ode:=(y(x)*(x^2+y(x)^2)^(1/2)+(y(x)^2-x^2)*sin(alpha)-2*x*y(x)*cos(alpha))*diff(y(x),x)+x*(x^2+y(x)^2)^(1/2)+2*x*y(x)*sin(alpha)+(y(x)^2-x^2)*cos(alpha) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \operatorname {RootOf}\left (-\ln \left (x \right )-\int _{}^{\textit {\_Z}}\frac {\textit {\_a}^{3} \cos \left (2 \alpha \right )+3 \textit {\_a}^{2} \sin \left (2 \alpha \right )+\textit {\_a}^{3}+\sqrt {2}\, \sqrt {\left (\textit {\_a}^{2}+1\right ) \left (\textit {\_a}^{2} \cos \left (2 \alpha \right )+2 \textit {\_a} \sin \left (2 \alpha \right )+\textit {\_a}^{2}-\cos \left (2 \alpha \right )+1\right )}-3 \textit {\_a} \cos \left (2 \alpha \right )-\sin \left (2 \alpha \right )+\textit {\_a}}{\left (\textit {\_a}^{2}+1\right ) \left (\textit {\_a}^{2} \cos \left (2 \alpha \right )+2 \textit {\_a} \sin \left (2 \alpha \right )+\textit {\_a}^{2}-\cos \left (2 \alpha \right )+1\right )}d \textit {\_a} +c_1 \right ) x \]

✓ Mathematica. Time used: 5.507 (sec). Leaf size: 116

ode=2*x*Sin[\[Alpha]]*y[x] + Cos[\[Alpha]]*(-x^2 + y[x]^2) + x*Sqrt[x^2 + y[x]^2] + (-2*x*Cos[\[Alpha]]*y[x] + Sin[\[Alpha]]*(-x^2 + y[x]^2) + y[x]*Sqrt[x^2 + y[x]^2])*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\sqrt {\cos ^2(\alpha )} \sec (\alpha ) \left (\log \left (\cos (\alpha ) \left (\sin (\alpha )+\frac {\cos (\alpha ) y(x)}{x}\right )\right )-\log \left (\frac {1}{2} \left (\cos (2 \alpha )-2 \sqrt {\cos ^2(\alpha )} \sqrt {\frac {y(x)^2}{x^2}+1}-\frac {\sin (2 \alpha ) y(x)}{x}+1\right )\right )\right )+\log \left (\frac {y(x)^2}{x^2}+1\right )-\frac {1}{2} \log \left (\left (\sin (\alpha )+\frac {\cos (\alpha ) y(x)}{x}\right )^2\right )=-\log (x)+c_1,y(x)\right ] \]

✗ Sympy

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

Timed Out