4.30.41 $ax{y}^{\prime }(x)+y(x)(b+c{x}^{2k})+x^2{y}^{″}(x)=0$

ODE
$$ax{y}^{\prime }(x)+y(x)(b+c{x}^{2k})+x^2{y}^{″}(x)=0$$ ODE Classification

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.0939198 (sec), leaf count = 561

$$\left\{\{y(x)\to 2^{-\frac{\sqrt{k^2(a^2-2a-4b+1)}+k\sqrt{a^2-2a-4b+1}-ak+k}{2k^2}}{k}^{-\frac{\sqrt{k^2(a^2-2a-4b+1)}+k\sqrt{a^2-2a-4b+1}-ak+k}{2k^2}}{c}^{-\frac{\sqrt{k^2(a^2-2a-4b+1)}+k(\sqrt{a^2-2a-4b+1}+a-1)}{4k^2}}{\left(x^{2k}\right)}^{-\frac{\sqrt{k^2(a^2-2a-4b+1)}+k(\sqrt{a^2-2a-4b+1}+a-1)}{4k^2}}(c_2{2}^{\frac{\sqrt{k^2(a^2-2a-4b+1)}}{k^2}}{k}^{\frac{\sqrt{k^2(a^2-2a-4b+1)}}{k^2}}{c}^{\frac{\sqrt{a^2-2a-4b+1}}{2k}}{\left(x^{2k}\right)}^{\frac{\sqrt{a^2-2a-4b+1}}{2k}}\mathrm{\Gamma }(\frac{\sqrt{a^2-2a-4b+1}}{2k}+1)J_{\frac{\sqrt{(a^2-2a-4b+1)k^2}}{2k^2}}\left(\frac{\sqrt{c}\sqrt{x^{2k}}}{k}\right)+c_1{2}^{\frac{\sqrt{a^2-2a-4b+1}}{k}}{k}^{\frac{\sqrt{a^2-2a-4b+1}}{k}}{c}^{\frac{\sqrt{k^2(a^2-2a-4b+1)}}{2k^2}}{\left(x^{2k}\right)}^{\frac{\sqrt{k^2(a^2-2a-4b+1)}}{2k^2}}\mathrm{\Gamma }(1-\frac{\sqrt{a^2-2a-4b+1}}{2k})J_{-\frac{\sqrt{(a^2-2a-4b+1)k^2}}{2k^2}}\left(\frac{\sqrt{c}\sqrt{x^{2k}}}{k}\right))\}\right\}$$

Maple ✓
cpu = 0.036 (sec), leaf count = 75

$$\{y\left(x\right)=x^{-\frac{a}{2}+\frac{1}{2}}({\mathit{Y}}_{\frac{1}{2k}\sqrt{a^2-2a-4b+1}}\left(\frac{x^k}{k}\sqrt{c}\right)\mathit{\_}\mathit{C}\mathit{2}+{\mathit{J}}_{\frac{1}{2k}\sqrt{a^2-2a-4b+1}}\left(\frac{x^k}{k}\sqrt{c}\right)\mathit{\_}\mathit{C}\mathit{1})\}$$ 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+Be
sselJ(1/2*(a^2-2*a-4*b+1)^(1/2)/k,c^(1/2)*x^k/k)*_C1)