4.33.38 $4(x^2+1)y^{″}(x)=x^2+4x{y}^{\prime }(x)$

ODE
$$4(x^2+1)y^{″}(x)=x^2+4x{y}^{\prime }(x)$$ ODE Classification

[[_2nd_order, _missing_y]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.0716702 (sec), leaf count = 57

$$\left\{\{y(x)\to \frac{1}{16}(8c_1\sqrt{x^2+1}x+8c_1{\sinh }^{-1}(x)+16c_2-3x^2+2\sqrt{x^2+1}x{\sinh }^{-1}(x)+{\sinh }^{-1}(x{)}^2)\}\right\}$$

Maple ✓
cpu = 0.117 (sec), leaf count = 39

$$\{y\left(x\right)=\frac{x}{2}(\frac{\mathit{A}\mathit{r}\mathit{c}\mathit{s}\mathit{i}\mathit{n}\mathit{h}\left(x\right)}{4}+\mathit{\_}\mathit{C}\mathit{1})\sqrt{x^2+1}-\frac{3x^2}{16}+\frac{\mathit{\_}\mathit{C}\mathit{1}\mathit{A}\mathit{r}\mathit{c}\mathit{s}\mathit{i}\mathit{n}\mathit{h}\left(x\right)}{2}+\frac{{\left(\mathit{A}\mathit{r}\mathit{c}\mathit{s}\mathit{i}\mathit{n}\mathit{h}\left(x\right)\right)}^2}{16}+\mathit{\_}\mathit{C}\mathit{2}-\frac{1}{16}\}$$ Mathematica raw input

DSolve[4*(1 + x^2)*y''[x] == x^2 + 4*x*y'[x],y[x],x]

Mathematica raw output

{{y[x] -> (-3*x^2 + 2*x*Sqrt[1 + x^2]*ArcSinh[x] + ArcSinh[x]^2 + 8*x*Sqrt[1 + x
^2]*C[1] + 8*ArcSinh[x]*C[1] + 16*C[2])/16}}

Maple raw input

dsolve(4*(x^2+1)*diff(diff(y(x),x),x) = 4*x*diff(y(x),x)+x^2, y(x),'implicit')

Maple raw output

y(x) = 1/2*x*(1/4*arcsinh(x)+_C1)*(x^2+1)^(1/2)-3/16*x^2+1/2*_C1*arcsinh(x)+1/16
*arcsinh(x)^2+_C2-1/16