ODE
\[ a x y'(x)+y(x) \left (b+c x^{2 k}\right )+x^2 y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0939198 (sec), leaf count = 561
\[\left \{\left \{y(x)\to 2^{-\frac {\sqrt {k^2 \left (a^2-2 a-4 b+1\right )}+k \sqrt {a^2-2 a-4 b+1}-a k+k}{2 k^2}} k^{-\frac {\sqrt {k^2 \left (a^2-2 a-4 b+1\right )}+k \sqrt {a^2-2 a-4 b+1}-a k+k}{2 k^2}} c^{-\frac {\sqrt {k^2 \left (a^2-2 a-4 b+1\right )}+k \left (\sqrt {a^2-2 a-4 b+1}+a-1\right )}{4 k^2}} \left (x^{2 k}\right )^{-\frac {\sqrt {k^2 \left (a^2-2 a-4 b+1\right )}+k \left (\sqrt {a^2-2 a-4 b+1}+a-1\right )}{4 k^2}} \left (c_2 2^{\frac {\sqrt {k^2 \left (a^2-2 a-4 b+1\right )}}{k^2}} k^{\frac {\sqrt {k^2 \left (a^2-2 a-4 b+1\right )}}{k^2}} c^{\frac {\sqrt {a^2-2 a-4 b+1}}{2 k}} \left (x^{2 k}\right )^{\frac {\sqrt {a^2-2 a-4 b+1}}{2 k}} \Gamma \left (\frac {\sqrt {a^2-2 a-4 b+1}}{2 k}+1\right ) J_{\frac {\sqrt {\left (a^2-2 a-4 b+1\right ) k^2}}{2 k^2}}\left (\frac {\sqrt {c} \sqrt {x^{2 k}}}{k}\right )+c_1 2^{\frac {\sqrt {a^2-2 a-4 b+1}}{k}} k^{\frac {\sqrt {a^2-2 a-4 b+1}}{k}} c^{\frac {\sqrt {k^2 \left (a^2-2 a-4 b+1\right )}}{2 k^2}} \left (x^{2 k}\right )^{\frac {\sqrt {k^2 \left (a^2-2 a-4 b+1\right )}}{2 k^2}} \Gamma \left (1-\frac {\sqrt {a^2-2 a-4 b+1}}{2 k}\right ) J_{-\frac {\sqrt {\left (a^2-2 a-4 b+1\right ) k^2}}{2 k^2}}\left (\frac {\sqrt {c} \sqrt {x^{2 k}}}{k}\right )\right )\right \}\right \}\]
Maple ✓
cpu = 0.036 (sec), leaf count = 75
\[ \left \{ y \left ( x \right ) ={x}^{-{\frac {a}{2}}+{\frac {1}{2}}} \left ( {{\sl Y}_{{\frac {1}{2\,k}\sqrt {{a}^{2}-2\,a-4\,b+1}}}\left ({\frac {{x}^{k}}{k}\sqrt {c}}\right )}{\it \_C2}+{{\sl J}_{{\frac {1}{2\,k}\sqrt {{a}^{2}-2\,a-4\,b+1}}}\left ({\frac {{x}^{k}}{k}\sqrt {c}}\right )}{\it \_C1} \right ) \right \} \] Mathematica raw input
DSolve[(b + c*x^(2*k))*y[x] + a*x*y'[x] + x^2*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (2^(Sqrt[1 - 2*a + a^2 - 4*b]/k)*c^(Sqrt[(1 - 2*a + a^2 - 4*b)*k^2]/(2*k^2))*k^(Sqrt[1 - 2*a + a^2 - 4*b]/k)*(x^(2*k))^(Sqrt[(1 - 2*a + a^2 - 4*b)*k^2]/(2*k^2))*BesselJ[-Sqrt[(1 - 2*a + a^2 - 4*b)*k^2]/(2*k^2), (Sqrt[c]*Sqrt[x^(2*k)])/k]*C[1]*Gamma[1 - Sqrt[1 - 2*a + a^2 - 4*b]/(2*k)] + 2^(Sqrt[(1 - 2*a + a^2 - 4*b)*k^2]/k^2)*c^(Sqrt[1 - 2*a + a^2 - 4*b]/(2*k))*k^(Sqrt[(1 - 2*a + a^2 - 4*b)*k^2]/k^2)*(x^(2*k))^(Sqrt[1 - 2*a + a^2 - 4*b]/(2*k))*BesselJ[Sqrt[(1 - 2*a + a^2 - 4*b)*k^2]/(2*k^2), (Sqrt[c]*Sqrt[x^(2*k)])/k]*C[2]*Gamma[1 + Sqrt[1 - 2*a + a^2 - 4*b]/(2*k)])/(2^((k - a*k + Sqrt[1 - 2*a + a^2 - 4*b]*k + Sqrt[(1 - 2*a + a^2 - 4*b)*k^2])/(2*k^2))*c^(((-1 + a + Sqrt[1 - 2*a + a^2 - 4*b])*k + Sqrt[(1 - 2*a + a^2 - 4*b)*k^2])/(4*k^2))*k^((k - a*k + Sqrt[1 - 2*a + a^2 - 4*b]*k + Sqrt[(1 - 2*a + a^2 - 4*b)*k^2])/(2*k^2))*(x^(2*k))^(((-1 + a + Sqrt[1 - 2*a + a^2 - 4*b])*k + Sqrt[(1 - 2*a + a^2 - 4*b)*k^2])/(4*k^2)))}}
Maple raw input
dsolve(x^2*diff(diff(y(x),x),x)+a*x*diff(y(x),x)+(b+c*x^(2*k))*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = x^(-1/2*a+1/2)*(BesselY(1/2*(a^2-2*a-4*b+1)^(1/2)/k,c^(1/2)*x^k/k)*_C2+BesselJ(1/2*(a^2-2*a-4*b+1)^(1/2)/k,c^(1/2)*x^k/k)*_C1)