Internal problem ID [8830]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 495.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, `class A`], _dAlembert]
\[ \boxed {\left (y^{2}+\left (-a +1\right ) x^{2}\right ) {y^{\prime }}^{2}+2 a x y y^{\prime }+\left (-a +1\right ) y^{2}=-x^{2}} \]
✓ Solution by Maple
Time used: 0.438 (sec). Leaf size: 75
dsolve((y(x)^2+(1-a)*x^2)*diff(y(x),x)^2+2*a*x*y(x)*diff(y(x),x)+(1-a)*y(x)^2+x^2 = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) = -i x y \left (x \right ) = i x y \left (x \right ) = \tan \left (\operatorname {RootOf}\left (-2 \textit {\_Z} \sqrt {a -1}-\ln \left (\frac {x^{2}}{\cos \left (\textit {\_Z} \right )^{2}}\right )+2 c_{1} \right )\right ) x y \left (x \right ) = \tan \left (\operatorname {RootOf}\left (2 \textit {\_Z} \sqrt {a -1}-\ln \left (\frac {x^{2}}{\cos \left (\textit {\_Z} \right )^{2}}\right )+2 c_{1} \right )\right ) x \end{align*}
✓ Solution by Mathematica
Time used: 0.319 (sec). Leaf size: 101
DSolve[x^2 + (1 - a)*y[x]^2 + 2*a*x*y[x]*y'[x] + ((1 - a)*x^2 + y[x]^2)*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\sqrt {a-1} \arctan \left (\frac {y(x)}{x}\right )-\frac {1}{2} \log \left (\frac {y(x)^2}{x^2}+1\right )=\log (x)+c_1,y(x)\right ] \text {Solve}\left [\sqrt {a-1} \arctan \left (\frac {y(x)}{x}\right )+\frac {1}{2} \log \left (\frac {y(x)^2}{x^2}+1\right )=-\log (x)+c_1,y(x)\right ] y(x)\to -i x y(x)\to i x \end{align*}