\[ y''(x)=-\frac {y'(x) (-\text {cn}(x|k) \text {dn}(x|k)-2 \text {sn}(x|k))}{\text {sn}(x|k)^2-\text {sn}(a|k)^2}-\frac {y(x) \left (6 k^2 \text {sn}(a|k)^4-4 \left (k^2+1\right ) \text {sn}(a|k)^2+2\right )}{\text {sn}(x|k)^2-\text {sn}(a|k)^2}-\frac {1}{\text {sn}(x|k)^2-\text {sn}(a|k)^2} \] ✗ Mathematica : cpu = 1.0719 (sec), leaf count = 0 , could not solve
DSolve[Derivative[2][y][x] == -(-JacobiSN[a, k]^2 + JacobiSN[x, k]^2)^(-1) - ((2 - 4*(1 + k^2)*JacobiSN[a, k]^2 + 6*k^2*JacobiSN[a, k]^4)*y[x])/(-JacobiSN[a, k]^2 + JacobiSN[x, k]^2) - ((-(JacobiCN[x, k]*JacobiDN[x, k]) - 2*JacobiSN[x, k])*Derivative[1][y][x])/(-JacobiSN[a, k]^2 + JacobiSN[x, k]^2), y[x], x]
✗ Maple : cpu = 0. (sec), leaf count = 0 , result contains DESol
\[ \left \{ y \left ( x \right ) ={\it DESol} \left ( \left \{ {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}{\it \_Y} \left ( x \right ) -2\,{\frac {{\it JacobiSN} \left ( x,k \right ) {\it JacobiCN} \left ( x,k \right ) {\it JacobiDN} \left ( x,k \right ) {\frac {\rm d}{{\rm d}x}}{\it \_Y} \left ( x \right ) }{ \left ( {\it JacobiSN} \left ( x,k \right ) \right ) ^{2}-{\it JacobiSN} \left ( a,k \right ) }}-{\frac { \left ( -2+4\, \left ( {k}^{2}+1 \right ) \left ( {\it JacobiSN} \left ( a,k \right ) \right ) ^{2}-6\,{k}^{2} \left ( {\it JacobiSN} \left ( a,k \right ) \right ) ^{4} \right ) {\it \_Y} \left ( x \right ) }{ \left ( {\it JacobiSN} \left ( x,k \right ) \right ) ^{2}-{\it JacobiSN} \left ( a,k \right ) }} \right \} , \left \{ {\it \_Y} \left ( x \right ) \right \} \right ) \right \} \]