ODE
\[ x^2 y''(x) \left (\text {a0}+\text {b0} x^k\right )+x y'(x) \left (\text {a1}+\text {b1} x^k\right )+y(x) \left (\text {a2}+\text {b2} x^k\right )=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.151316 (sec), leaf count = 540
\[\left \{\left \{y(x)\to \text {a0}^{-\frac {\sqrt {\text {a0}^2-2 \text {a0} (\text {a1}+2 \text {a2})+\text {a1}^2}+\text {a0}-\text {a1}}{2 \text {a0} k}} \text {b0}^{-\frac {\sqrt {\text {a0}^2-2 \text {a0} (\text {a1}+2 \text {a2})+\text {a1}^2}-\text {a0}+\text {a1}}{2 \text {a0} k}} \left (x^k\right )^{-\frac {\sqrt {\text {a0}^2-2 \text {a0} (\text {a1}+2 \text {a2})+\text {a1}^2}-\text {a0}+\text {a1}}{2 \text {a0} k}} \left (c_2 \text {b0}^{\frac {\sqrt {\text {a0}^2-2 \text {a0} (\text {a1}+2 \text {a2})+\text {a1}^2}}{\text {a0} k}} \left (x^k\right )^{\frac {\sqrt {\text {a0}^2-2 \text {a0} (\text {a1}+2 \text {a2})+\text {a1}^2}}{\text {a0} k}} \, _2F_1\left (\frac {\text {b1} \text {a0}-\sqrt {\text {b0}^2-2 (\text {b1}+2 \text {b2}) \text {b0}+\text {b1}^2} \text {a0}-\text {a1} \text {b0}+\sqrt {\text {a0}^2-2 (\text {a1}+2 \text {a2}) \text {a0}+\text {a1}^2} \text {b0}}{2 \text {a0} \text {b0} k},\frac {\text {b1} \text {a0}+\sqrt {\text {b0}^2-2 (\text {b1}+2 \text {b2}) \text {b0}+\text {b1}^2} \text {a0}-\text {a1} \text {b0}+\sqrt {\text {a0}^2-2 (\text {a1}+2 \text {a2}) \text {a0}+\text {a1}^2} \text {b0}}{2 \text {a0} \text {b0} k};\frac {\sqrt {\text {a0}^2-2 (\text {a1}+2 \text {a2}) \text {a0}+\text {a1}^2}}{\text {a0} k}+1;-\frac {\text {b0} x^k}{\text {a0}}\right )+c_1 \text {a0}^{\frac {\sqrt {\text {a0}^2-2 \text {a0} (\text {a1}+2 \text {a2})+\text {a1}^2}}{\text {a0} k}} \, _2F_1\left (-\frac {-\text {b1} \text {a0}+\sqrt {\text {b0}^2-2 (\text {b1}+2 \text {b2}) \text {b0}+\text {b1}^2} \text {a0}+\text {a1} \text {b0}+\sqrt {\text {a0}^2-2 (\text {a1}+2 \text {a2}) \text {a0}+\text {a1}^2} \text {b0}}{2 \text {a0} \text {b0} k},\frac {\text {b1} \text {a0}+\sqrt {\text {b0}^2-2 (\text {b1}+2 \text {b2}) \text {b0}+\text {b1}^2} \text {a0}-\text {a1} \text {b0}-\sqrt {\text {a0}^2-2 (\text {a1}+2 \text {a2}) \text {a0}+\text {a1}^2} \text {b0}}{2 \text {a0} \text {b0} k};1-\frac {\sqrt {\text {a0}^2-2 (\text {a1}+2 \text {a2}) \text {a0}+\text {a1}^2}}{\text {a0} k};-\frac {\text {b0} x^k}{\text {a0}}\right )\right )\right \}\right \}\]
Maple ✓
cpu = 0.392 (sec), leaf count = 391
\[ \left \{ y \left ( x \right ) ={\it \_C1}\,{\mbox {$_2$F$_1$}({\frac {1}{2\,{\it b0}\,{\it a0}\,k} \left ( \sqrt {{{\it a0}}^{2}+ \left ( -2\,{\it a1}-4\,{\it a2} \right ) {\it a0}+{{\it a1}}^{2}}{\it b0}+\sqrt {{{\it b0}}^{2}+ \left ( -2\,{\it b1}-4\,{\it b2} \right ) {\it b0}+{{\it b1}}^{2}}{\it a0}+{\it b1}\,{\it a0}-{\it a1}\,{\it b0} \right ) },{\frac {1}{2\,{\it b0}\,{\it a0}\,k} \left ( \sqrt {{{\it a0}}^{2}+ \left ( -2\,{\it a1}-4\,{\it a2} \right ) {\it a0}+{{\it a1}}^{2}}{\it b0}-\sqrt {{{\it b0}}^{2}+ \left ( -2\,{\it b1}-4\,{\it b2} \right ) {\it b0}+{{\it b1}}^{2}}{\it a0}+{\it b1}\,{\it a0}-{\it a1}\,{\it b0} \right ) };\,{\frac {1}{k{\it a0}} \left ( k{\it a0}+\sqrt {{{\it a0}}^{2}+ \left ( -2\,{\it a1}-4\,{\it a2} \right ) {\it a0}+{{\it a1}}^{2}} \right ) };\,-{\frac {{\it b0}\,{x}^{k}}{{\it a0}}})}{x}^{{\frac {1}{2\,{\it a0}} \left ( -{\it a1}+\sqrt {{{\it a0}}^{2}+ \left ( -2\,{\it a1}-4\,{\it a2} \right ) {\it a0}+{{\it a1}}^{2}}+{\it a0} \right ) }}+{\it \_C2}\,{\mbox {$_2$F$_1$}(-{\frac {1}{2\,{\it b0}\,{\it a0}\,k} \left ( \sqrt {{{\it a0}}^{2}+ \left ( -2\,{\it a1}-4\,{\it a2} \right ) {\it a0}+{{\it a1}}^{2}}{\it b0}-\sqrt {{{\it b0}}^{2}+ \left ( -2\,{\it b1}-4\,{\it b2} \right ) {\it b0}+{{\it b1}}^{2}}{\it a0}-{\it b1}\,{\it a0}+{\it a1}\,{\it b0} \right ) },-{\frac {1}{2\,{\it b0}\,{\it a0}\,k} \left ( \sqrt {{{\it a0}}^{2}+ \left ( -2\,{\it a1}-4\,{\it a2} \right ) {\it a0}+{{\it a1}}^{2}}{\it b0}+\sqrt {{{\it b0}}^{2}+ \left ( -2\,{\it b1}-4\,{\it b2} \right ) {\it b0}+{{\it b1}}^{2}}{\it a0}-{\it b1}\,{\it a0}+{\it a1}\,{\it b0} \right ) };\,{\frac {1}{k{\it a0}} \left ( k{\it a0}-\sqrt {{{\it a0}}^{2}+ \left ( -2\,{\it a1}-4\,{\it a2} \right ) {\it a0}+{{\it a1}}^{2}} \right ) };\,-{\frac {{\it b0}\,{x}^{k}}{{\it a0}}})}{x}^{-{\frac {1}{2\,{\it a0}} \left ( {\it a1}+\sqrt {{{\it a0}}^{2}+ \left ( -2\,{\it a1}-4\,{\it a2} \right ) {\it a0}+{{\it a1}}^{2}}-{\it a0} \right ) }} \right \} \] Mathematica raw input
DSolve[(a2 + b2*x^k)*y[x] + x*(a1 + b1*x^k)*y'[x] + x^2*(a0 + b0*x^k)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (b0^(Sqrt[a0^2 + a1^2 - 2*a0*(a1 + 2*a2)]/(a0*k))*(x^k)^(Sqrt[a0^2 + a1^2 - 2*a0*(a1 + 2*a2)]/(a0*k))*C[2]*Hypergeometric2F1[(-(a1*b0) + Sqrt[a0^2 + a1^2 - 2*a0*(a1 + 2*a2)]*b0 + a0*b1 - a0*Sqrt[b0^2 + b1^2 - 2*b0*(b1 + 2*b2)])/(2*a0*b0*k), (-(a1*b0) + Sqrt[a0^2 + a1^2 - 2*a0*(a1 + 2*a2)]*b0 + a0*b1 + a0*Sqrt[b0^2 + b1^2 - 2*b0*(b1 + 2*b2)])/(2*a0*b0*k), 1 + Sqrt[a0^2 + a1^2 - 2*a0*(a1 + 2*a2)]/(a0*k), -((b0*x^k)/a0)] + a0^(Sqrt[a0^2 + a1^2 - 2*a0*(a1 + 2*a2)]/(a0*k))*C[1]*Hypergeometric2F1[-(a1*b0 + Sqrt[a0^2 + a1^2 - 2*a0*(a1 + 2*a2)]*b0 - a0*b1 + a0*Sqrt[b0^2 + b1^2 - 2*b0*(b1 + 2*b2)])/(2*a0*b0*k), (-(a1*b0) - Sqrt[a0^2 + a1^2 - 2*a0*(a1 + 2*a2)]*b0 + a0*b1 + a0*Sqrt[b0^2 + b1^2 - 2*b0*(b1 + 2*b2)])/(2*a0*b0*k), 1 - Sqrt[a0^2 + a1^2 - 2*a0*(a1 + 2*a2)]/(a0*k), -((b0*x^k)/a0)])/(a0^((a0 - a1 + Sqrt[a0^2 + a1^2 - 2*a0*(a1 + 2*a2)])/(2*a0*k))*b0^((-a0 + a1 + Sqrt[a0^2 + a1^2 - 2*a0*(a1 + 2*a2)])/(2*a0*k))*(x^k)^((-a0 + a1 + Sqrt[a0^2 + a1^2 - 2*a0*(a1 + 2*a2)])/(2*a0*k)))}}
Maple raw input
dsolve(x^2*(a0+b0*x^k)*diff(diff(y(x),x),x)+x*(a1+b1*x^k)*diff(y(x),x)+(a2+b2*x^k)*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = _C1*hypergeom([1/2/b0/a0/k*((a0^2+(-2*a1-4*a2)*a0+a1^2)^(1/2)*b0+(b0^2+(-2*b1-4*b2)*b0+b1^2)^(1/2)*a0+b1*a0-a1*b0), 1/2/b0/a0/k*((a0^2+(-2*a1-4*a2)*a0+a1^2)^(1/2)*b0-(b0^2+(-2*b1-4*b2)*b0+b1^2)^(1/2)*a0+b1*a0-a1*b0)],[(k*a0+(a0^2+(-2*a1-4*a2)*a0+a1^2)^(1/2))/k/a0],-1/a0*b0*x^k)*x^(1/2*(-a1+(a0^2+(-2*a1-4*a2)*a0+a1^2)^(1/2)+a0)/a0)+_C2*hypergeom([-1/2/b0/a0/k*((a0^2+(-2*a1-4*a2)*a0+a1^2)^(1/2)*b0-(b0^2+(-2*b1-4*b2)*b0+b1^2)^(1/2)*a0-b1*a0+a1*b0), -1/2/b0/a0/k*((a0^2+(-2*a1-4*a2)*a0+a1^2)^(1/2)*b0+(b0^2+(-2*b1-4*b2)*b0+b1^2)^(1/2)*a0-b1*a0+a1*b0)],[(k*a0-(a0^2+(-2*a1-4*a2)*a0+a1^2)^(1/2))/k/a0],-1/a0*b0*x^k)*x^(-1/2*(a1+(a0^2+(-2*a1-4*a2)*a0+a1^2)^(1/2)-a0)/a0)