4.35.46 $(a-x{)}^2(b-x{)}^2{y}^{″}(x)=k^{2}y(x)$

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

$$\left\{\{y(x)\to (x-a{)}^{\frac{1}{2}(1-\sqrt{\frac{4k^2}{(a-b{)}^2}+1})}(x-b{)}^{\frac{1}{2}(1-\sqrt{\frac{4k^2}{(a-b{)}^2}+1})}(c_1(x-a{)}^{\sqrt{\frac{4k^2}{(a-b{)}^2}+1}}-\frac{c_2(x-b{)}^{\sqrt{\frac{4k^2}{(a-b{)}^2}+1}}}{(a-b)\sqrt{\frac{4k^2}{(a-b{)}^2}+1}})\}\right\}$$

Maple ✓
cpu = 0.132 (sec), leaf count = 108

$$\{y\left(x\right)=\sqrt{(a-x)(b-x)}({\left(\frac{a-x}{b-x}\right)}^{\frac{1}{2a-2b}\sqrt{a^2-2ab+b^2+4k^2}}\mathit{\_}\mathit{C}\mathit{1}+{\left(\frac{a-x}{b-x}\right)}^{-\frac{1}{2a-2b}\sqrt{a^2-2ab+b^2+4k^2}}\mathit{\_}\mathit{C}\mathit{2})\}$$ 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)^S
qrt[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)