4.19.1 \(a^2+2 a x y'(x)-2 a y(x)+x^2 y'(x)^2+x^2=0\)

ODE
\[ a^2+2 a x y'(x)-2 a y(x)+x^2 y'(x)^2+x^2=0 \] ODE Classification

[_rational]

Book solution method
No Missing Variables ODE, Solve for \(y\)

Mathematica ✓
cpu = 1.46917 (sec), leaf count = 54

\[\text {Solve}\left [\left \{\frac {x^2 \left (K[1]^2+1\right )}{a}+2 x K[1]+a=2 y(x),x=\frac {-a \sinh ^{-1}(K[1])+c_1}{\sqrt {K[1]^2+1}}\right \},\{y(x),K[1]\}\right ]\]

Maple ✓
cpu = 16.655 (sec), leaf count = 236

\[\left [y \left (x \right ) = \frac {\RootOf \left (\arcsinh \left (\textit {\_Z} \right )^{2} a^{2}-x^{2} \textit {\_Z}^{2}-2 \arcsinh \left (\textit {\_Z} \right ) \textit {\_C1} a +\textit {\_C1}^{2}-x^{2}\right )^{2} x^{2}}{2 a}-\frac {a \arcsinh \left (\RootOf \left (\arcsinh \left (\textit {\_Z} \right )^{2} a^{2}-x^{2} \textit {\_Z}^{2}-2 \arcsinh \left (\textit {\_Z} \right ) \textit {\_C1} a +\textit {\_C1}^{2}-x^{2}\right )\right ) \RootOf \left (\arcsinh \left (\textit {\_Z} \right )^{2} a^{2}-x^{2} \textit {\_Z}^{2}-2 \arcsinh \left (\textit {\_Z} \right ) \textit {\_C1} a +\textit {\_C1}^{2}-x^{2}\right )}{\sqrt {\RootOf \left (\arcsinh \left (\textit {\_Z} \right )^{2} a^{2}-x^{2} \textit {\_Z}^{2}-2 \arcsinh \left (\textit {\_Z} \right ) \textit {\_C1} a +\textit {\_C1}^{2}-x^{2}\right )^{2}+1}}+\frac {a}{2}+\frac {x^{2}}{2 a}+\frac {\textit {\_C1} \RootOf \left (\arcsinh \left (\textit {\_Z} \right )^{2} a^{2}-x^{2} \textit {\_Z}^{2}-2 \arcsinh \left (\textit {\_Z} \right ) \textit {\_C1} a +\textit {\_C1}^{2}-x^{2}\right )}{\sqrt {\RootOf \left (\arcsinh \left (\textit {\_Z} \right )^{2} a^{2}-x^{2} \textit {\_Z}^{2}-2 \arcsinh \left (\textit {\_Z} \right ) \textit {\_C1} a +\textit {\_C1}^{2}-x^{2}\right )^{2}+1}}\right ]\] Mathematica raw input

DSolve[a^2 + x^2 - 2*a*y[x] + 2*a*x*y'[x] + x^2*y'[x]^2 == 0,y[x],x]

Mathematica raw output

Solve[{a + 2*x*K[1] + (x^2*(1 + K[1]^2))/a == 2*y[x], x == (-(a*ArcSinh[K[1]]) +
 C[1])/Sqrt[1 + K[1]^2]}, {y[x], K[1]}]

Maple raw input

dsolve(x^2*diff(y(x),x)^2+2*a*x*diff(y(x),x)+a^2+x^2-2*a*y(x) = 0, y(x))

Maple raw output

[y(x) = 1/2/a*RootOf(arcsinh(_Z)^2*a^2-x^2*_Z^2-2*arcsinh(_Z)*_C1*a+_C1^2-x^2)^2
*x^2-a/(RootOf(arcsinh(_Z)^2*a^2-x^2*_Z^2-2*arcsinh(_Z)*_C1*a+_C1^2-x^2)^2+1)^(1
/2)*arcsinh(RootOf(arcsinh(_Z)^2*a^2-x^2*_Z^2-2*arcsinh(_Z)*_C1*a+_C1^2-x^2))*Ro
otOf(arcsinh(_Z)^2*a^2-x^2*_Z^2-2*arcsinh(_Z)*_C1*a+_C1^2-x^2)+1/2*a+1/2/a*x^2+1
/(RootOf(arcsinh(_Z)^2*a^2-x^2*_Z^2-2*arcsinh(_Z)*_C1*a+_C1^2-x^2)^2+1)^(1/2)*_C
1*RootOf(arcsinh(_Z)^2*a^2-x^2*_Z^2-2*arcsinh(_Z)*_C1*a+_C1^2-x^2)]