4.35.38 ${(a^2+x^2)}^2(b^2+x^2)y^{″}(x)+x(\text{a0}+\text{b0}{x}^2)y^{\prime }(x)+y(x)(\text{a1}+\text{b1}{x}^2)=0$

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

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✗
cpu = 139.621 (sec), leaf count = 0 , DifferentialRoot result

$$\text{left {left {y(x)to text {DifferentialRoot}left ({unicode {f818},unicode {f817}}unicode {f4a1}left {left (unicode {f817}^2+b^2right ) unicode {f818}''(unicode {f817}) left (unicode {f817}^2+a^2right )^2+left (text {b1} unicode {f817}^2+text {a1}right ) unicode {f818}(unicode {f817})+left (text {b0} unicode {f817}^3+text {a0} unicode {f817}right ) unicode {f818}'(unicode {f817})=0,unicode {f818}(0)=c\_1,unicode {f818}'(0)=c\_2right }right )(x)right }right }}$$

Maple ✗
cpu = 1.707 (sec), leaf count = 0 , result contains DESol

$$\{y\left(x\right)=\mathit{D}\mathit{E}\mathit{S}\mathit{o}\mathit{l}(\{\frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}\mathit{\_}\mathit{Y}\left(x\right)+\frac{x(x^2\mathit{b}\mathit{0}+\mathit{a}\mathit{0})\frac{\mathrm{d}}{\mathrm{d}x}\mathit{\_}\mathit{Y}\left(x\right)}{{(a^2+x^2)}^2(b^2+x^2)}+\frac{(\mathit{b}\mathit{1}{x}^2+\mathit{a}\mathit{1})\mathit{\_}\mathit{Y}\left(x\right)}{{(a^2+x^2)}^2(b^2+x^2)}\},\left\{\mathit{\_}\mathit{Y}\left(x\right)\right\})\}$$

Mathematica raw input

DSolve[(a1 + b1*x^2)*y[x] + x*(a0 + b0*x^2)*y'[x] + (a^2 + x^2)^2*(b^2 + x^2)*y''[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> DifferentialRoot[Function[{\[FormalY], \[FormalX]}, {(a1 + \[FormalX]^
2*b1)*\[FormalY][\[FormalX]] + (\[FormalX]*a0 + \[FormalX]^3*b0)*Derivative[1][\
[FormalY]][\[FormalX]] + (\[FormalX]^2 + a^2)^2*(\[FormalX]^2 + b^2)*Derivative[
2][\[FormalY]][\[FormalX]] == 0, \[FormalY][0] == C[1], Derivative[1][\[FormalY]
][0] == C[2]}]][x]}}

Maple raw input

dsolve((a^2+x^2)^2*(b^2+x^2)*diff(diff(y(x),x),x)+x*(b0*x^2+a0)*diff(y(x),x)+(b1*x^2+a1)*y(x) = 0, y(x),'implicit')

Maple raw output

y(x) = DESol({diff(diff(_Y(x),x),x)+x*(b0*x^2+a0)/(a^2+x^2)^2/(b^2+x^2)*diff(_Y(
x),x)+(b1*x^2+a1)/(a^2+x^2)^2/(b^2+x^2)*_Y(x)},{_Y(x)})