60.1.337 problem 338

Internal problem ID [10351]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 338
Date solved : Monday, January 27, 2025 at 07:19:13 PM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

✓ Solution by Maple

Time used: 3.582 (sec). Leaf size: 128

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)
 

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

✓ Solution by Mathematica

Time used: 5.679 (sec). Leaf size: 85

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])*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {\sin (\alpha ) K[1]^2-2 \cos (\alpha ) K[1]+\sqrt {K[1]^2+1} K[1]-\sin (\alpha )}{\left (K[1]^2+1\right ) \left (-\cos (\alpha )+K[1] \sin (\alpha )+\sqrt {K[1]^2+1}\right )}dK[1]=-\log (x)+c_1,y(x)\right ] \]