ODE
\[ 4 x^2 y''(x)-\left (4 x^2+1\right ) y(x)+4 x y'(x)=4 e^x x^{3/2} \] ODE Classification
[[_2nd_order, _linear, _nonhomogeneous]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.175673 (sec), leaf count = 39
\[\left \{\left \{y(x)\to \frac {e^{-x} \left (e^{2 x} (2 x-1+2 c_2)+4 c_1\right )}{4 \sqrt {x}}\right \}\right \}\]
Maple ✓
cpu = 0.279 (sec), leaf count = 26
\[\left [y \left (x \right ) = \frac {\sinh \left (x \right ) \textit {\_C2}}{\sqrt {x}}+\frac {\cosh \left (x \right ) \textit {\_C1}}{\sqrt {x}}+\frac {{\mathrm e}^{x} \sqrt {x}}{2}\right ]\] Mathematica raw input
DSolve[-((1 + 4*x^2)*y[x]) + 4*x*y'[x] + 4*x^2*y''[x] == 4*E^x*x^(3/2),y[x],x]
Mathematica raw output
{{y[x] -> (4*C[1] + E^(2*x)*(-1 + 2*x + 2*C[2]))/(4*E^x*Sqrt[x])}}
Maple raw input
dsolve(4*x^2*diff(diff(y(x),x),x)+4*x*diff(y(x),x)-(4*x^2+1)*y(x) = 4*exp(x)*x^(3/2), y(x))
Maple raw output
[y(x) = sinh(x)/x^(1/2)*_C2+cosh(x)/x^(1/2)*_C1+1/2*exp(x)*x^(1/2)]