Internal problem ID [7918]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 338.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class A], _dAlembert]
Solve \begin {gather*} \boxed {\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 {gather*}
✓ Solution by Maple
Time used: 1.281 (sec). Leaf size: 127
dsolve((y(x)*(y(x)^2+x^2)^(1/2)+(y(x)^2-x^2)*sin(alpha)-2*x*y(x)*cos(alpha))*diff(y(x),x)+x*(y(x)^2+x^2)^(1/2)+2*x*y(x)*sin(alpha)+(y(x)^2-x^2)*cos(alpha) = 0,y(x), singsol=all)
\[ y \relax (x ) = \RootOf \left (-\ln \relax (x )+\int _{}^{\textit {\_Z}}-\frac {\textit {\_a}^{3} \cos \left (2 \alpha \right )+3 \textit {\_a}^{2} \sin \left (2 \alpha \right )+\textit {\_a}^{3}-3 \textit {\_a} \cos \left (2 \alpha \right )-\sin \left (2 \alpha \right )+\sqrt {2}\, \sqrt {\left (\textit {\_a}^{2}+1\right ) \left (\textit {\_a}^{2} \cos \left (2 \alpha \right )+2 \textit {\_a} \sin \left (2 \alpha \right )+\textit {\_a}^{2}-\cos \left (2 \alpha \right )+1\right )}+\textit {\_a}}{\left (\textit {\_a}^{2}+1\right ) \left (\textit {\_a}^{2} \cos \left (2 \alpha \right )+2 \textit {\_a} \sin \left (2 \alpha \right )+\textit {\_a}^{2}-\cos \left (2 \alpha \right )+1\right )}d \textit {\_a} +c_{1}\right ) x \]
✓ Solution by Mathematica
Time used: 4.205 (sec). Leaf size: 116
DSolve[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])*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\sqrt {\cos ^2(\alpha )} \sec (\alpha ) \left (\log \left (\cos (\alpha ) \left (\sin (\alpha )+\frac {\cos (\alpha ) y(x)}{x}\right )\right )-\log \left (\frac {1}{2} \left (\cos (2 \alpha )-2 \sqrt {\cos ^2(\alpha )} \sqrt {\frac {y(x)^2}{x^2}+1}-\frac {\sin (2 \alpha ) y(x)}{x}+1\right )\right )\right )+\log \left (\frac {y(x)^2}{x^2}+1\right )-\frac {1}{2} \log \left (\left (\sin (\alpha )+\frac {\cos (\alpha ) y(x)}{x}\right )^2\right )=-\log (x)+c_1,y(x)\right ] \]