ODE
\[ (a-x)^2 (b-x)^2 y''(x)=k^2 y(x) \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.772337 (sec), leaf count = 151
\[\left \{\left \{y(x)\to (x-a)^{\frac {1}{2} \left (1-\sqrt {\frac {4 k^2}{(a-b)^2}+1}\right )} (x-b)^{\frac {1}{2} \left (1-\sqrt {\frac {4 k^2}{(a-b)^2}+1}\right )} \left (c_1 (x-a)^{\sqrt {\frac {4 k^2}{(a-b)^2}+1}}-\frac {c_2 (x-b)^{\sqrt {\frac {4 k^2}{(a-b)^2}+1}}}{(a-b) \sqrt {\frac {4 k^2}{(a-b)^2}+1}}\right )\right \}\right \}\]
Maple ✓
cpu = 0.273 (sec), leaf count = 120
\[\left [y \left (x \right ) = \textit {\_C1} \left (\frac {a -x}{b -x}\right )^{\frac {\sqrt {a^{2}-2 a b +b^{2}+4 k^{2}}}{2 a -2 b}} \sqrt {\left (a -x \right ) \left (b -x \right )}+\textit {\_C2} \left (\frac {a -x}{b -x}\right )^{-\frac {\sqrt {a^{2}-2 a b +b^{2}+4 k^{2}}}{2 a -2 b}} \sqrt {\left (a -x \right ) \left (b -x \right )}\right ]\] Mathematica raw input
DSolve[(a - x)^2*(b - x)^2*y''[x] == k^2*y[x],y[x],x]
Mathematica raw output
{{y[x] -> (-a + x)^((1 - Sqrt[1 + (4*k^2)/(a - b)^2])/2)*(-b + x)^((1 - Sqrt[1 + (4*k^2)/(a - b)^2])/2)*((-a + x)^Sqrt[1 + (4*k^2)/(a - b)^2]*C[1] - ((-b + x)^Sqrt[1 + (4*k^2)/(a - b)^2]*C[2])/((a - b)*Sqrt[1 + (4*k^2)/(a - b)^2]))}}
Maple raw input
dsolve((a-x)^2*(b-x)^2*diff(diff(y(x),x),x) = k^2*y(x), y(x))
Maple raw output
[y(x) = _C1*(1/(b-x)*(a-x))^((a^2-2*a*b+b^2+4*k^2)^(1/2)/(2*a-2*b))*((a-x)*(b-x))^(1/2)+_C2*(1/(b-x)*(a-x))^(-(a^2-2*a*b+b^2+4*k^2)^(1/2)/(2*a-2*b))*((a-x)*(b-x))^(1/2)]