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 = 1.7483 (sec), leaf count = 54

\[\text {Solve}\left [\left \{y(x)=\frac {a^2+2 a \text {K$\$$1542706} x+\left (\text {K$\$$1542706}^2+1\right ) x^2}{2 a},x=\frac {c_1-a \sinh ^{-1}(\text {K$\$$1542706})}{\sqrt {\text {K$\$$1542706}^2+1}}\right \},\{y(x),\text {K$\$$1542706}\}\right ]\]

Maple
cpu = 10.321 (sec), leaf count = 80

\[ \left \{ [y \left ({\it \_T} \right ) =-{\frac {1}{a} \left (\left (-{\frac {{a}^{2} \left ({\it Arcsinh} \left ({\it \_T} \right ) \right ) ^{2}}{2}}+a{\it \_C1}\,{\it Arcsinh} \left ({\it \_T} \right ) -{\frac {{a}^{2}}{2}}-{\frac {{{\it \_C1}}^{2}}{2}} \right ) \sqrt {{{\it \_T}}^{2}+1}+a{\it \_T}\, \left (a{\it Arcsinh} \left ({\it \_T} \right ) -{\it \_C1} \right ) \right ) {\frac {1}{\sqrt {{{\it \_T}}^{2}+1}}}},x \left ({\it \_T} \right ) ={(-a{\it Arcsinh} \left ({\it \_T} \right ) +{\it \_C1}){\frac {1}{\sqrt {{{\it \_T}}^{2}+1}}}}] \right \} \] 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[{y[x] == (a^2 + 2*a*K$1542706*x + (1 + K$1542706^2)*x^2)/(2*a), x == (-(a*
ArcSinh[K$1542706]) + C[1])/Sqrt[1 + K$1542706^2]}, {y[x], K$1542706}]

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),'implicit')

Maple raw output

[y(_T) = -1/(_T^2+1)^(1/2)*((-1/2*a^2*arcsinh(_T)^2+a*_C1*arcsinh(_T)-1/2*a^2-1/
2*_C1^2)*(_T^2+1)^(1/2)+a*_T*(a*arcsinh(_T)-_C1))/a, x(_T) = (-a*arcsinh(_T)+_C1
)/(_T^2+1)^(1/2)]