4.44.14 $x^2{y}^{‴}(x)+a{x}^{2}y(x)-6y^{\prime }(x)=0$

ODE
$$x^2{y}^{‴}(x)+a{x}^{2}y(x)-6y^{\prime }(x)=0$$ ODE Classification

[[_3rd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.553604 (sec), leaf count = 97

$$\left\{\{y(x)\to \frac{c_1{e}^{-\sqrt[3]{a}x}(\sqrt[3]{a}x+2)+c_2{e}^{\sqrt[3]{-1}\sqrt[3]{a}x}(\sqrt[3]{a}x+2(-1{)}^{2/3})+c_3{e}^{-(-1{)}^{2/3}\sqrt[3]{a}x}(\sqrt[3]{a}x-2\sqrt[3]{-1})}{x}\}\right\}$$

Maple ✓
cpu = 0.556 (sec), leaf count = 135

$$\{y\left(x\right)=\frac{1}{x}(-\mathit{\_}\mathit{C}\mathit{2}((-i-\sqrt{3}){(-a^4)}^{\frac{2}{3}}+i{a}^{3}x){\mathrm{e}}^{\frac{\frac{i}{2}(-\sqrt{3}+i)x}{a}\sqrt[3]{-a^4}}-\mathit{\_}\mathit{C}\mathit{3}((-i+\sqrt{3}){(-a^4)}^{\frac{2}{3}}+i{a}^{3}x){\mathrm{e}}^{\frac{\frac{i}{2}(\sqrt{3}+i)x}{a}\sqrt[3]{-a^4}}+\mathit{\_}\mathit{C}\mathit{1}{\mathrm{e}}^{\frac{x}{a}\sqrt[3]{-a^4}}(a^{3}x+2{(-a^4)}^{2/3}))\}$$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> (((2 + a^(1/3)*x)*C[1])/E^(a^(1/3)*x) + E^((-1)^(1/3)*a^(1/3)*x)*(2*(-
1)^(2/3) + a^(1/3)*x)*C[2] + ((-2*(-1)^(1/3) + a^(1/3)*x)*C[3])/E^((-1)^(2/3)*a^
(1/3)*x))/x}}

Maple raw input

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

Maple raw output

y(x) = (-_C2*((-I-3^(1/2))*(-a^4)^(2/3)+I*a^3*x)*exp(1/2*I*(-3^(1/2)+I)*(-a^4)^(
1/3)*x/a)-_C3*((-I+3^(1/2))*(-a^4)^(2/3)+I*a^3*x)*exp(1/2*I*(3^(1/2)+I)*(-a^4)^(
1/3)*x/a)+_C1*exp((-a^4)^(1/3)/a*x)*(a^3*x+2*(-a^4)^(2/3)))/x