Internal problem ID [4219]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 33
Problem number: 987.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, `class A`], _rational, _dAlembert]
\[ \boxed {\left (\left (-a^{2}+1\right ) x^{2}+y^{2}\right ) {y^{\prime }}^{2}+2 a^{2} x y y^{\prime }+\left (-a^{2}+1\right ) y^{2}=-x^{2}} \]
✓ Solution by Maple
Time used: 0.157 (sec). Leaf size: 79
dsolve(((-a^2+1)*x^2+y(x)^2)*diff(y(x),x)^2+2*a^2*x*y(x)*diff(y(x),x)+x^2+(-a^2+1)*y(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^{2}-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^{2}-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: 0.381 (sec). Leaf size: 115
DSolve[((1-a^2)x^2+y[x]^2) (y'[x])^2 +2 a^2 x y[x] y'[x]+x^2+(1-a^2) y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\sqrt {a-1} \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} \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*}