ODE
\[ x^2 (a-x)^2 y''(x)+b y(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.319527 (sec), leaf count = 121
\[\left \{\left \{y(x)\to \frac {x^{\frac {1}{2}-\frac {1}{2} \sqrt {1-\frac {4 b}{a^2}}} (x-a)^{\frac {1}{2}-\frac {1}{2} \sqrt {1-\frac {4 b}{a^2}}} \left (a c_1 \sqrt {1-\frac {4 b}{a^2}} x^{\sqrt {1-\frac {4 b}{a^2}}}+c_2 (x-a)^{\sqrt {1-\frac {4 b}{a^2}}}\right )}{a \sqrt {1-\frac {4 b}{a^2}}}\right \}\right \}\]
Maple ✓
cpu = 0.087 (sec), leaf count = 67
\[ \left \{ y \left ( x \right ) =\sqrt {x \left ( a-x \right ) } \left ( \left ( {\frac {x}{a-x}} \right ) ^{{\frac {1}{2\,a}\sqrt {{a}^{2}-4\,b}}}{\it \_C2}+ \left ( {\frac {a-x}{x}} \right ) ^{{\frac {1}{2\,a}\sqrt {{a}^{2}-4\,b}}}{\it \_C1} \right ) \right \} \] Mathematica raw input
DSolve[b*y[x] + (a - x)^2*x^2*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (x^(1/2 - Sqrt[1 - (4*b)/a^2]/2)*(-a + x)^(1/2 - Sqrt[1 - (4*b)/a^2]/2)*(a*Sqrt[1 - (4*b)/a^2]*x^Sqrt[1 - (4*b)/a^2]*C[1] + (-a + x)^Sqrt[1 - (4*b)/a^2]*C[2]))/(a*Sqrt[1 - (4*b)/a^2])}}
Maple raw input
dsolve(x^2*(a-x)^2*diff(diff(y(x),x),x)+b*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = (x*(a-x))^(1/2)*((x/(a-x))^(1/2*(a^2-4*b)^(1/2)/a)*_C2+(1/x*(a-x))^(1/2*(a^2-4*b)^(1/2)/a)*_C1)