4.33.2 $-4(a+x)y^{\prime }(x)+(\text{a0}+x{)}^2{y}^{″}(x)+6y(x)=0$

ODE
$$-4(a+x)y^{\prime }(x)+(\text{a0}+x{)}^2{y}^{″}(x)+6y(x)=0$$ ODE Classification

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.475826 (sec), leaf count = 179

$$\left\{\{y(x)\to \frac{(8a^2-4a(\text{a0}-3x)-{\text{a0}}^2-6\text{a0}x+3x^2)(\frac{1}{9}{c}_2(e^{-\frac{4(a-\text{a0})}{\text{a0}+x}}(x-\frac{8(a-\text{a0}{)}^2(6a+\text{a0}+7x)}{8a^2-4a(\text{a0}-3x)-{\text{a0}}^2-6\text{a0}x+3x^2})+12a\text{Ei}(-\frac{4(a-\text{a0})}{\text{a0}+x})-12\text{a0}\text{Ei}(-\frac{4(a-\text{a0})}{\text{a0}+x})+(6a-5\text{a0})e^{-\frac{4(a-\text{a0})}{\text{a0}+x}})+c_1)}{8a^2-4a\text{a0}-{\text{a0}}^2}\}\right\}$$

Maple ✓
cpu = 0.127 (sec), leaf count = 132

$$\{y\left(x\right)=-32(-1/8{\mathit{a}\mathit{0}}^2+(-a/2-3/4x)\mathit{a}\mathit{0}+a^2+3/2ax+3/8x^2)(a-\mathit{a}\mathit{0})\mathit{\_}\mathit{C}\mathit{2}\mathit{E}\mathit{i}(1,\frac{4a-4\mathit{a}\mathit{0}}{\mathit{a}\mathit{0}+x})+8(\mathit{a}\mathit{0}+x)\mathit{\_}\mathit{C}\mathit{2}(-1/8{\mathit{a}\mathit{0}}^2+(-3/4a-x)\mathit{a}\mathit{0}+a^2+5/4ax+1/8x^2){\mathrm{e}}^{\frac{-4a+4\mathit{a}\mathit{0}}{\mathit{a}\mathit{0}+x}}+\frac{8\mathit{\_}\mathit{C}\mathit{1}}{3}(-\frac{{\mathit{a}\mathit{0}}^2}{8}+(-\frac{a}{2}-\frac{3x}{4})\mathit{a}\mathit{0}+a^2+\frac{3ax}{2}+\frac{3x^2}{8})\}$$ Mathematica raw input

DSolve[6*y[x] - 4*(a + x)*y'[x] + (a0 + x)^2*y''[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> ((8*a^2 - a0^2 - 4*a*(a0 - 3*x) - 6*a0*x + 3*x^2)*(C[1] + (C[2]*((6*a 
- 5*a0)/E^((4*(a - a0))/(a0 + x)) + (x - (8*(a - a0)^2*(6*a + a0 + 7*x))/(8*a^2 
- a0^2 - 4*a*(a0 - 3*x) - 6*a0*x + 3*x^2))/E^((4*(a - a0))/(a0 + x)) + 12*a*ExpI
ntegralEi[(-4*(a - a0))/(a0 + x)] - 12*a0*ExpIntegralEi[(-4*(a - a0))/(a0 + x)])
)/9))/(8*a^2 - 4*a*a0 - a0^2)}}

Maple raw input

dsolve((a0+x)^2*diff(diff(y(x),x),x)-4*(a+x)*diff(y(x),x)+6*y(x) = 0, y(x),'implicit')

Maple raw output

y(x) = -32*(-1/8*a0^2+(-1/2*a-3/4*x)*a0+a^2+3/2*a*x+3/8*x^2)*(a-a0)*_C2*Ei(1,(4*
a-4*a0)/(a0+x))+8*(a0+x)*_C2*(-1/8*a0^2+(-3/4*a-x)*a0+a^2+5/4*a*x+1/8*x^2)*exp((
-4*a+4*a0)/(a0+x))+8/3*_C1*(-1/8*a0^2+(-1/2*a-3/4*x)*a0+a^2+3/2*a*x+3/8*x^2)