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 = 0.220501 (sec), leaf count = 154
\[\left \{\left \{y(x)\to \frac {-\sqrt {x^2-1} x-9 c_2 \sqrt {1-x^2} x^2+9 c_1 \sqrt {-\left (x^2-1\right )^2} x+9 c_2 \sqrt {1-x^2}-9 c_2 \left (x^2-1\right ) x \sin ^{-1}(x)-4 \sqrt {x^2-1} x^5+\left (5 \sqrt {x^2-1}-3 \sqrt {1-x^2} \sqrt {-\left (x^2-1\right )^2}\right ) x^3}{9 \sqrt {-\left (x^2-1\right )^2}}\right \}\right \}\]
Maple ✓
cpu = 0.126 (sec), leaf count = 50
\[\left [y \left (x \right ) = x \textit {\_C2} +\left (\ln \left (x +\sqrt {x^{2}-1}\right ) x -\sqrt {x^{2}-1}\right ) \textit {\_C1} +\frac {\left (x^{3}-x \right ) \sqrt {-x^{2}+1}}{9}\right ]\] 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] -> (-(x*Sqrt[-1 + x^2]) - 4*x^5*Sqrt[-1 + x^2] + x^3*(5*Sqrt[-1 + x^2] - 3*Sqrt[1 - x^2]*Sqrt[-(-1 + x^2)^2]) + 9*x*Sqrt[-(-1 + x^2)^2]*C[1] + 9*Sqrt[1 - x^2]*C[2] - 9*x^2*Sqrt[1 - x^2]*C[2] - 9*x*(-1 + x^2)*ArcSin[x]*C[2])/(9*Sqrt[-(-1 + x^2)^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))
Maple raw output
[y(x) = x*_C2+(ln(x+(x^2-1)^(1/2))*x-(x^2-1)^(1/2))*_C1+1/9*(x^3-x)*(-x^2+1)^(1/2)]