ODE
\[ n (n+1) y(x)+\left (1-x^2\right ) y''(x)-2 x y'(x)=\frac {2 ((-n-1) x P_n(x)+(n+1) P_{n+1}(x))}{x^2-1} \] ODE Classification
[[_2nd_order, _linear, _nonhomogeneous]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.808737 (sec), leaf count = 159
\[\left \{\left \{y(x)\to P_n(x) \int _1^x\frac {2 (K[1] P_n(K[1])-P_{n+1}(K[1])) Q_n(K[1])}{(K[1]-1) (K[1]+1) (P_{n+1}(K[1]) Q_n(K[1])-P_n(K[1]) Q_{n+1}(K[1]))}dK[1]+Q_n(x) \int _1^x\frac {2 P_n(K[2]) (P_{n+1}(K[2])-K[2] P_n(K[2]))}{(K[2]-1) (K[2]+1) (P_{n+1}(K[2]) Q_n(K[2])-P_n(K[2]) Q_{n+1}(K[2]))}dK[2]+c_1 P_n(x)+c_2 Q_n(x)\right \}\right \}\]
Maple ✓
cpu = 0.638 (sec), leaf count = 119
\[\left [y \left (x \right ) = \LegendreP \left (n , x\right ) \textit {\_C2} +\LegendreQ \left (n , x\right ) \textit {\_C1} -2 \left (\int \frac {\LegendreQ \left (n , x\right ) \left (-x \LegendreP \left (n , x\right )+\LegendreP \left (n +1, x\right )\right )}{\left (\LegendreP \left (n +1, x\right ) \LegendreQ \left (n , x\right )-\LegendreP \left (n , x\right ) \LegendreQ \left (n +1, x\right )\right ) \left (x^{2}-1\right )}d x \right ) \LegendreP \left (n , x\right )+2 \left (\int \frac {\left (-x \LegendreP \left (n , x\right )+\LegendreP \left (n +1, x\right )\right ) \LegendreP \left (n , x\right )}{\left (\LegendreP \left (n +1, x\right ) \LegendreQ \left (n , x\right )-\LegendreP \left (n , x\right ) \LegendreQ \left (n +1, x\right )\right ) \left (x^{2}-1\right )}d x \right ) \LegendreQ \left (n , x\right )\right ]\] Mathematica raw input
DSolve[n*(1 + n)*y[x] - 2*x*y'[x] + (1 - x^2)*y''[x] == (2*((-1 - n)*x*LegendreP[n, x] + (1 + n)*LegendreP[1 + n, x]))/(-1 + x^2),y[x],x]
Mathematica raw output
{{y[x] -> C[1]*LegendreP[n, x] + C[2]*LegendreQ[n, x] + LegendreP[n, x]*Inactive[Integrate][(2*(K[1]*LegendreP[n, K[1]] - LegendreP[1 + n, K[1]])*LegendreQ[n, K[1]])/((-1 + K[1])*(1 + K[1])*(LegendreP[1 + n, K[1]]*LegendreQ[n, K[1]] - LegendreP[n, K[1]]*LegendreQ[1 + n, K[1]])), {K[1], 1, x}] + LegendreQ[n, x]*Inactive[Integrate][(2*LegendreP[n, K[2]]*(-(K[2]*LegendreP[n, K[2]]) + LegendreP[1 + n, K[2]]))/((-1 + K[2])*(1 + K[2])*(LegendreP[1 + n, K[2]]*LegendreQ[n, K[2]] - LegendreP[n, K[2]]*LegendreQ[1 + n, K[2]])), {K[2], 1, x}]}}
Maple raw input
dsolve((-x^2+1)*diff(diff(y(x),x),x)-2*x*diff(y(x),x)+n*(n+1)*y(x) = 2*((n+1)*LegendreP(n+1,x)-(n+1)*x*LegendreP(n,x))/(x^2-1), y(x))
Maple raw output
[y(x) = LegendreP(n,x)*_C2+LegendreQ(n,x)*_C1-2*Int(LegendreQ(n,x)*(-x*LegendreP(n,x)+LegendreP(n+1,x))/(LegendreP(n+1,x)*LegendreQ(n,x)-LegendreP(n,x)*LegendreQ(n+1,x))/(x^2-1),x)*LegendreP(n,x)+2*Int((-x*LegendreP(n,x)+LegendreP(n+1,x))*LegendreP(n,x)/(LegendreP(n+1,x)*LegendreQ(n,x)-LegendreP(n,x)*LegendreQ(n+1,x))/(x^2-1),x)*LegendreQ(n,x)]