ODE
\[ (a+b x)^4 y''(x)+y(x)=0 \] ODE Classification
[[_Emden, _Fowler]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.170791 (sec), leaf count = 76
\[\left \{\left \{y(x)\to \frac {1}{2} e^{-\frac {\sqrt {-\frac {1}{b^4}} b}{a+b x}} (a+b x) \left (2 c_1 e^{\frac {2 \sqrt {-\frac {1}{b^4}} b}{a+b x}}+\left (-\frac {1}{b^4}\right )^{3/2} b^6 c_2\right )\right \}\right \}\]
Maple ✓
cpu = 0.08 (sec), leaf count = 39
\[ \left \{ y \left ( x \right ) = \left ( bx+a \right ) \left ( {\it \_C2}\,\cos \left ( {\frac {1}{b \left ( bx+a \right ) }} \right ) +{\it \_C1}\,\sin \left ( {\frac {1}{b \left ( bx+a \right ) }} \right ) \right ) \right \} \] Mathematica raw input
DSolve[y[x] + (a + b*x)^4*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> ((a + b*x)*(2*E^((2*Sqrt[-b^(-4)]*b)/(a + b*x))*C[1] + (-b^(-4))^(3/2)*b^6*C[2]))/(2*E^((Sqrt[-b^(-4)]*b)/(a + b*x)))}}
Maple raw input
dsolve((b*x+a)^4*diff(diff(y(x),x),x)+y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = (b*x+a)*(_C2*cos(1/b/(b*x+a))+_C1*sin(1/b/(b*x+a)))