ODE
\[ \left (1-x^2\right ) y''(x)+x y'(x)-y(x)=x \left (1-x^2\right )^{3/2} \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.108009 (sec), leaf count = 173
\[\left \{\left \{y(x)\to \frac {\left (x^2-1\right ) \left (-9 c_1 \sqrt {1-x^2} x+9 c_2 \sqrt {-\left (x^2-1\right )^2}+x^5-2 x^3+x\right )-3 \left (-3 c_2 \sqrt {1-x^2} x+3 c_2 \sqrt {1-x^2} x^3+\sqrt {x^2-1} x^6-\left (\sqrt {x^2-1}-\sqrt {1-x^2} \sqrt {-\left (x^2-1\right )^2}\right ) x^4\right ) \log \left (\sqrt {x^2-1}+x\right )}{9 \left (1-x^2\right )^{3/2}}\right \}\right \}\]
Maple ✓
cpu = 0.053 (sec), leaf count = 49
\[ \left \{ y \left ( x \right ) ={\frac {{x}^{3}-x}{9}\sqrt {-{x}^{2}+1}}+\ln \left ( x+\sqrt {{x}^{2}-1} \right ) {\it \_C1}\,x+x{\it \_C2}-\sqrt {{x}^{2}-1}{\it \_C1} \right \} \] Mathematica raw input
DSolve[-y[x] + x*y'[x] + (1 - x^2)*y''[x] == x*(1 - x^2)^(3/2),y[x],x]
Mathematica raw output
{{y[x] -> ((-1 + x^2)*(x - 2*x^3 + x^5 - 9*x*Sqrt[1 - x^2]*C[1] + 9*Sqrt[-(-1 + x^2)^2]*C[2]) - 3*(x^6*Sqrt[-1 + x^2] - x^4*(Sqrt[-1 + x^2] - Sqrt[1 - x^2]*Sqrt[-(-1 + x^2)^2]) - 3*x*Sqrt[1 - x^2]*C[2] + 3*x^3*Sqrt[1 - x^2]*C[2])*Log[x + Sqrt[-1 + x^2]])/(9*(1 - x^2)^(3/2))}}
Maple raw input
dsolve((-x^2+1)*diff(diff(y(x),x),x)+x*diff(y(x),x)-y(x) = x*(-x^2+1)^(3/2), y(x),'implicit')
Maple raw output
y(x) = 1/9*(x^3-x)*(-x^2+1)^(1/2)+ln(x+(x^2-1)^(1/2))*_C1*x+x*_C2-(x^2-1)^(1/2)*_C1