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 = 0.715449 (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 = 0.132 (sec), leaf count = 108
\[ \left \{ y \left ( x \right ) =\sqrt { \left ( a-x \right ) \left ( b-x \right ) } \left ( \left ( {\frac {a-x}{b-x}} \right ) ^{{\frac {1}{2\,a-2\,b}\sqrt {{a}^{2}-2\,ab+{b}^{2}+4\,{k}^{2}}}}{\it \_C1}+ \left ( {\frac {a-x}{b-x}} \right ) ^{-{\frac {1}{2\,a-2\,b}\sqrt {{a}^{2}-2\,ab+{b}^{2}+4\,{k}^{2}}}}{\it \_C2} \right ) \right \} \] 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),'implicit')
Maple raw output
y(x) = ((a-x)*(b-x))^(1/2)*((1/(b-x)*(a-x))^((a^2-2*a*b+b^2+4*k^2)^(1/2)/(2*a-2*b))*_C1+(1/(b-x)*(a-x))^(-(a^2-2*a*b+b^2+4*k^2)^(1/2)/(2*a-2*b))*_C2)