ODE
\[ x y'(x) \left (\text {a0}+\text {b0} x^k\right )+y(x) \left (\text {a1}+\text {b1} x^k+\text {c1} x^{2 k}\right )+x^2 y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.124977 (sec), leaf count = 414
\[\left \{\left \{y(x)\to 2^{\frac {1}{2} \left (\frac {\sqrt {k^2 \left (\text {a0}^2-2 \text {a0}-4 \text {a1}+1\right )}}{k^2}+1\right )} x^{\frac {1}{2} (-\text {a0}-k+1)} \left (x^k\right )^{\frac {1}{2} \left (\frac {\sqrt {k^2 \left (\text {a0}^2-2 \text {a0}-4 \text {a1}+1\right )}}{k^2}+1\right )} e^{-\frac {\left (\sqrt {\text {b0}^2-4 \text {c1}}+\text {b0}\right ) x^k}{2 k}} \left (c_1 U\left (\frac {\left (k^2+\sqrt {\left (\text {a0}^2-2 \text {a0}-4 \text {a1}+1\right ) k^2}\right ) \text {b0}^2+\sqrt {\text {b0}^2-4 \text {c1}} k (\text {a0}+k-1) \text {b0}-2 \left (\text {b1} \sqrt {\text {b0}^2-4 \text {c1}} k+2 \text {c1} \left (k^2+\sqrt {\left (\text {a0}^2-2 \text {a0}-4 \text {a1}+1\right ) k^2}\right )\right )}{2 \left (\text {b0}^2-4 \text {c1}\right ) k^2},\frac {k^2+\sqrt {\left (\text {a0}^2-2 \text {a0}-4 \text {a1}+1\right ) k^2}}{k^2},\frac {\sqrt {\text {b0}^2-4 \text {c1}} x^k}{k}\right )+c_2 L_{-\frac {-2 \left (2 \text {c1} \left (\sqrt {k^2 \left (\text {a0}^2-2 \text {a0}-4 \text {a1}+1\right )}+k^2\right )+\text {b1} k \sqrt {\text {b0}^2-4 \text {c1}}\right )+\text {b0}^2 \left (\sqrt {k^2 \left (\text {a0}^2-2 \text {a0}-4 \text {a1}+1\right )}+k^2\right )+\text {b0} k (\text {a0}+k-1) \sqrt {\text {b0}^2-4 \text {c1}}}{2 k^2 \left (\text {b0}^2-4 \text {c1}\right )}}^{\frac {\sqrt {k^2 \left (\text {a0}^2-2 \text {a0}-4 \text {a1}+1\right )}}{k^2}}\left (\frac {\sqrt {\text {b0}^2-4 \text {c1}} x^k}{k}\right )\right )\right \}\right \}\]
Maple ✓
cpu = 0.222 (sec), leaf count = 148
\[ \left \{ y \left ( x \right ) ={x}^{-{\frac {{\it a0}}{2}}+{\frac {1}{2}}-{\frac {k}{2}}}{{\rm e}^{-{\frac {{\it b0}\,{x}^{k}}{2\,k}}}} \left ( {{\sl W}_{-{\frac { \left ( {\it a0}-1+k \right ) {\it b0}-2\,{\it b1}}{2\,k}{\frac {1}{\sqrt {{{\it b0}}^{2}-4\,{\it c1}}}}},\,{\frac {1}{2\,k}\sqrt {{{\it a0}}^{2}-2\,{\it a0}-4\,{\it a1}+1}}}\left ({\frac {{x}^{k}}{k}\sqrt {{{\it b0}}^{2}-4\,{\it c1}}}\right )}{\it \_C2}+{{\sl M}_{-{\frac { \left ( {\it a0}-1+k \right ) {\it b0}-2\,{\it b1}}{2\,k}{\frac {1}{\sqrt {{{\it b0}}^{2}-4\,{\it c1}}}}},\,{\frac {1}{2\,k}\sqrt {{{\it a0}}^{2}-2\,{\it a0}-4\,{\it a1}+1}}}\left ({\frac {{x}^{k}}{k}\sqrt {{{\it b0}}^{2}-4\,{\it c1}}}\right )}{\it \_C1} \right ) \right \} \] Mathematica raw input
DSolve[(a1 + b1*x^k + c1*x^(2*k))*y[x] + x*(a0 + b0*x^k)*y'[x] + x^2*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (2^((1 + Sqrt[(1 - 2*a0 + a0^2 - 4*a1)*k^2]/k^2)/2)*x^((1 - a0 - k)/2)*(x^k)^((1 + Sqrt[(1 - 2*a0 + a0^2 - 4*a1)*k^2]/k^2)/2)*(C[1]*HypergeometricU[(b0*Sqrt[b0^2 - 4*c1]*k*(-1 + a0 + k) + b0^2*(k^2 + Sqrt[(1 - 2*a0 + a0^2 - 4*a1)*k^2]) - 2*(b1*Sqrt[b0^2 - 4*c1]*k + 2*c1*(k^2 + Sqrt[(1 - 2*a0 + a0^2 - 4*a1)*k^2])))/(2*(b0^2 - 4*c1)*k^2), (k^2 + Sqrt[(1 - 2*a0 + a0^2 - 4*a1)*k^2])/k^2, (Sqrt[b0^2 - 4*c1]*x^k)/k] + C[2]*LaguerreL[-(b0*Sqrt[b0^2 - 4*c1]*k*(-1 + a0 + k) + b0^2*(k^2 + Sqrt[(1 - 2*a0 + a0^2 - 4*a1)*k^2]) - 2*(b1*Sqrt[b0^2 - 4*c1]*k + 2*c1*(k^2 + Sqrt[(1 - 2*a0 + a0^2 - 4*a1)*k^2])))/(2*(b0^2 - 4*c1)*k^2), Sqrt[(1 - 2*a0 + a0^2 - 4*a1)*k^2]/k^2, (Sqrt[b0^2 - 4*c1]*x^k)/k]))/E^(((b0 + Sqrt[b0^2 - 4*c1])*x^k)/(2*k))}}
Maple raw input
dsolve(x^2*diff(diff(y(x),x),x)+x*(a0+b0*x^k)*diff(y(x),x)+(a1+b1*x^k+c1*x^(2*k))*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = x^(-1/2*a0+1/2-1/2*k)*exp(-1/2/k*b0*x^k)*(WhittakerW(-1/2/(b0^2-4*c1)^(1/2)*((a0-1+k)*b0-2*b1)/k,1/2*(a0^2-2*a0-4*a1+1)^(1/2)/k,(b0^2-4*c1)^(1/2)/k*x^k)*_C2+WhittakerM(-1/2/(b0^2-4*c1)^(1/2)*((a0-1+k)*b0-2*b1)/k,1/2*(a0^2-2*a0-4*a1+1)^(1/2)/k,(b0^2-4*c1)^(1/2)/k*x^k)*_C1)