4.7.41 \(a x^2 y'(x)=a x y(x)+b^2 y(x)^2+x^2\)

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

[[_homogeneous, `class A`], _rational, _Riccati]

Book solution method
Homogeneous equation

Mathematica
cpu = 0.0190003 (sec), leaf count = 23

\[\left \{\left \{y(x)\to \frac {x \tan \left (\frac {b \left (a c_1+\log (x)\right )}{a}\right )}{b}\right \}\right \}\]

Maple
cpu = 0.012 (sec), leaf count = 24

\[ \left \{ {\frac {a}{b}\arctan \left ({\frac {by \relax (x ) }{x}} \right ) }-\ln \relax (x ) -{\it \_C1}=0 \right \} \] Mathematica raw input

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

Mathematica raw output

{{y[x] -> (x*Tan[(b*(a*C[1] + Log[x]))/a])/b}}

Maple raw input

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

Maple raw output

a/b*arctan(b/x*y(x))-ln(x)-_C1 = 0