4.31.12 $x{y}^{\prime }(x)(\text{a0}+\text{b0}{x}^k)+y(x)(\text{a1}+\text{b1}{x}^k+\text{c1}{x}^{2k})+x^2{y}^{″}(x)=0$

ODE
$$x{y}^{\prime }(x)(\text{a0}+\text{b0}{x}^k)+y(x)(\text{a1}+\text{b1}{x}^k+\text{c1}{x}^{2k})+x^2{y}^{″}(x)=0$$ ODE Classification

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.124977 (sec), leaf count = 414

$$\left\{\{y(x)\to 2^{\frac{1}{2}(\frac{\sqrt{k^2({\text{a0}}^2-2\text{a0}-4\text{a1}+1)}}{k^2}+1)}{x}^{\frac{1}{2}(-\text{a0}-k+1)}{\left(x^k\right)}^{\frac{1}{2}(\frac{\sqrt{k^2({\text{a0}}^2-2\text{a0}-4\text{a1}+1)}}{k^2}+1)}{e}^{-\frac{(\sqrt{{\text{b0}}^2-4\text{c1}}+\text{b0})x^k}{2k}}(c_{1}U(\frac{(k^2+\sqrt{({\text{a0}}^2-2\text{a0}-4\text{a1}+1)k^2}){\text{b0}}^2+\sqrt{{\text{b0}}^2-4\text{c1}}k(\text{a0}+k-1)\text{b0}-2(\text{b1}\sqrt{{\text{b0}}^2-4\text{c1}}k+2\text{c1}(k^2+\sqrt{({\text{a0}}^2-2\text{a0}-4\text{a1}+1)k^2}))}{2({\text{b0}}^2-4\text{c1})k^2},\frac{k^2+\sqrt{({\text{a0}}^2-2\text{a0}-4\text{a1}+1)k^2}}{k^2},\frac{\sqrt{{\text{b0}}^2-4\text{c1}}{x}^k}{k})+c_2{L}_{-\frac{-2(2\text{c1}(\sqrt{k^2({\text{a0}}^2-2\text{a0}-4\text{a1}+1)}+k^2)+\text{b1}k\sqrt{{\text{b0}}^2-4\text{c1}})+{\text{b0}}^2(\sqrt{k^2({\text{a0}}^2-2\text{a0}-4\text{a1}+1)}+k^2)+\text{b0}k(\text{a0}+k-1)\sqrt{{\text{b0}}^2-4\text{c1}}}{2k^2({\text{b0}}^2-4\text{c1})}}^{\frac{\sqrt{k^2({\text{a0}}^2-2\text{a0}-4\text{a1}+1)}}{k^2}}\left(\frac{\sqrt{{\text{b0}}^2-4\text{c1}}{x}^k}{k}\right))\}\right\}$$

Maple ✓
cpu = 0.222 (sec), leaf count = 148

$$\{y\left(x\right)=x^{-\frac{\mathit{a}\mathit{0}}{2}+\frac{1}{2}-\frac{k}{2}}{\mathrm{e}}^{-\frac{\mathit{b}\mathit{0}{x}^k}{2k}}({\mathit{W}}_{-\frac{(\mathit{a}\mathit{0}-1+k)\mathit{b}\mathit{0}-2\mathit{b}\mathit{1}}{2k}\frac{1}{\sqrt{{\mathit{b}\mathit{0}}^2-4\mathit{c}\mathit{1}}},\frac{1}{2k}\sqrt{{\mathit{a}\mathit{0}}^2-2\mathit{a}\mathit{0}-4\mathit{a}\mathit{1}+1}}\left(\frac{x^k}{k}\sqrt{{\mathit{b}\mathit{0}}^2-4\mathit{c}\mathit{1}}\right)\mathit{\_}\mathit{C}\mathit{2}+{\mathit{M}}_{-\frac{(\mathit{a}\mathit{0}-1+k)\mathit{b}\mathit{0}-2\mathit{b}\mathit{1}}{2k}\frac{1}{\sqrt{{\mathit{b}\mathit{0}}^2-4\mathit{c}\mathit{1}}},\frac{1}{2k}\sqrt{{\mathit{a}\mathit{0}}^2-2\mathit{a}\mathit{0}-4\mathit{a}\mathit{1}+1}}\left(\frac{x^k}{k}\sqrt{{\mathit{b}\mathit{0}}^2-4\mathit{c}\mathit{1}}\right)\mathit{\_}\mathit{C}\mathit{1})\}$$ 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[(b
0*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, (Sq
rt[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)