4.5.1 \(x y'(x)=a \sqrt {b^2 x^2+y(x)^2}+y(x)\)

ODE
\[ x y'(x)=a \sqrt {b^2 x^2+y(x)^2}+y(x) \] ODE Classification

[[_homogeneous, `class A`], _dAlembert]

Book solution method
Homogeneous equation

Mathematica
cpu = 0.045988 (sec), leaf count = 39

\[\left \{\left \{y(x)\to \frac {1}{2} e^{c_1} x^{a+1}-\frac {1}{2} b^2 e^{-c_1} x^{1-a}\right \}\right \}\]

Maple
cpu = 0.046 (sec), leaf count = 41

\[ \left \{ {\frac {y \relax (x ) }{{x}^{a}x}}+{\frac {1}{{x}^{a}x}\sqrt { \left (y \relax (x ) \right ) ^{2}+{b}^{2}{x}^{2}}}-{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[x*y'[x] == y[x] + a*Sqrt[b^2*x^2 + y[x]^2],y[x],x]

Mathematica raw output

{{y[x] -> -(b^2*x^(1 - a))/(2*E^C[1]) + (E^C[1]*x^(1 + a))/2}}

Maple raw input

dsolve(x*diff(y(x),x) = y(x)+a*(y(x)^2+b^2*x^2)^(1/2), y(x),'implicit')

Maple raw output

1/(x^a)/x*y(x)+1/(x^a)/x*(y(x)^2+b^2*x^2)^(1/2)-_C1 = 0