4.43.45 $y^{‴}(x)+\text{a1}{y}^{″}(x)+\text{a2}{y}^{\prime }(x)+\text{a3}y(x)=0$

ODE
$$y^{‴}(x)+\text{a1}{y}^{″}(x)+\text{a2}{y}^{\prime }(x)+\text{a3}y(x)=0$$ ODE Classification

[[_3rd_order, _missing_x]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.009094 (sec), leaf count = 84

$$\left\{\{y(x)\to c_1{e}^{x\text{Root}[{\mathrm{\#}\text{1}}^3+{\mathrm{\#}\text{1}}^2\text{a1}+\mathrm{\#}\text{1}\text{a2}+\text{a3}\mathrm{\&},1]}+c_2{e}^{x\text{Root}[{\mathrm{\#}\text{1}}^3+{\mathrm{\#}\text{1}}^2\text{a1}+\mathrm{\#}\text{1}\text{a2}+\text{a3}\mathrm{\&},2]}+c_3{e}^{x\text{Root}[{\mathrm{\#}\text{1}}^3+{\mathrm{\#}\text{1}}^2\text{a1}+\mathrm{\#}\text{1}\text{a2}+\text{a3}\mathrm{\&},3]}\}\right\}$$

Maple ✓
cpu = 0.02 (sec), leaf count = 542

$$\{y\left(x\right)=\mathit{\_}\mathit{C}\mathit{3}{\mathrm{e}}^{\frac{x}{6}({(36\mathit{a}\mathit{2}\mathit{a}\mathit{1}-108\mathit{a}\mathit{3}-8{\mathit{a}\mathit{1}}^3+12\sqrt{81{\mathit{a}\mathit{3}}^2+(12{\mathit{a}\mathit{1}}^3-54\mathit{a}\mathit{2}\mathit{a}\mathit{1})\mathit{a}\mathit{3}-3{\mathit{a}\mathit{2}}^2{\mathit{a}\mathit{1}}^2+12{\mathit{a}\mathit{2}}^3})}^{\frac{2}{3}}-2\mathit{a}\mathit{1}\sqrt[3]{36\mathit{a}\mathit{2}\mathit{a}\mathit{1}-108\mathit{a}\mathit{3}-8{\mathit{a}\mathit{1}}^3+12\sqrt{81{\mathit{a}\mathit{3}}^2+(12{\mathit{a}\mathit{1}}^3-54\mathit{a}\mathit{2}\mathit{a}\mathit{1})\mathit{a}\mathit{3}-3{\mathit{a}\mathit{2}}^2{\mathit{a}\mathit{1}}^2+12{\mathit{a}\mathit{2}}^3}}+4{\mathit{a}\mathit{1}}^2-12\mathit{a}\mathit{2})\frac{1}{\sqrt[3]{36\mathit{a}\mathit{2}\mathit{a}\mathit{1}-108\mathit{a}\mathit{3}-8{\mathit{a}\mathit{1}}^3+12\sqrt{81{\mathit{a}\mathit{3}}^2+(12{\mathit{a}\mathit{1}}^3-54\mathit{a}\mathit{2}\mathit{a}\mathit{1})\mathit{a}\mathit{3}-3{\mathit{a}\mathit{2}}^2{\mathit{a}\mathit{1}}^2+12{\mathit{a}\mathit{2}}^3}}}}+\mathit{\_}\mathit{C}\mathit{2}{\mathrm{e}}^{-\frac{x}{3}((-\frac{i}{4}\sqrt{3}+\frac{1}{4}){(36\mathit{a}\mathit{2}\mathit{a}\mathit{1}-108\mathit{a}\mathit{3}-8{\mathit{a}\mathit{1}}^3+12\sqrt{81{\mathit{a}\mathit{3}}^2+(12{\mathit{a}\mathit{1}}^3-54\mathit{a}\mathit{2}\mathit{a}\mathit{1})\mathit{a}\mathit{3}-3{\mathit{a}\mathit{2}}^2{\mathit{a}\mathit{1}}^2+12{\mathit{a}\mathit{2}}^3})}^{\frac{2}{3}}+\mathit{a}\mathit{1}\sqrt[3]{36\mathit{a}\mathit{2}\mathit{a}\mathit{1}-108\mathit{a}\mathit{3}-8{\mathit{a}\mathit{1}}^3+12\sqrt{81{\mathit{a}\mathit{3}}^2+(12{\mathit{a}\mathit{1}}^3-54\mathit{a}\mathit{2}\mathit{a}\mathit{1})\mathit{a}\mathit{3}-3{\mathit{a}\mathit{2}}^2{\mathit{a}\mathit{1}}^2+12{\mathit{a}\mathit{2}}^3}}+({\mathit{a}\mathit{1}}^2-3\mathit{a}\mathit{2})(i\sqrt{3}+1))\frac{1}{\sqrt[3]{36\mathit{a}\mathit{2}\mathit{a}\mathit{1}-108\mathit{a}\mathit{3}-8{\mathit{a}\mathit{1}}^3+12\sqrt{81{\mathit{a}\mathit{3}}^2+(12{\mathit{a}\mathit{1}}^3-54\mathit{a}\mathit{2}\mathit{a}\mathit{1})\mathit{a}\mathit{3}-3{\mathit{a}\mathit{2}}^2{\mathit{a}\mathit{1}}^2+12{\mathit{a}\mathit{2}}^3}}}}+\mathit{\_}\mathit{C}\mathit{1}{\mathrm{e}}^{\frac{x}{3}((-\frac{i}{4}\sqrt{3}-\frac{1}{4}){(36\mathit{a}\mathit{2}\mathit{a}\mathit{1}-108\mathit{a}\mathit{3}-8{\mathit{a}\mathit{1}}^3+12\sqrt{81{\mathit{a}\mathit{3}}^2+(12{\mathit{a}\mathit{1}}^3-54\mathit{a}\mathit{2}\mathit{a}\mathit{1})\mathit{a}\mathit{3}-3{\mathit{a}\mathit{2}}^2{\mathit{a}\mathit{1}}^2+12{\mathit{a}\mathit{2}}^3})}^{\frac{2}{3}}-\mathit{a}\mathit{1}\sqrt[3]{36\mathit{a}\mathit{2}\mathit{a}\mathit{1}-108\mathit{a}\mathit{3}-8{\mathit{a}\mathit{1}}^3+12\sqrt{81{\mathit{a}\mathit{3}}^2+(12{\mathit{a}\mathit{1}}^3-54\mathit{a}\mathit{2}\mathit{a}\mathit{1})\mathit{a}\mathit{3}-3{\mathit{a}\mathit{2}}^2{\mathit{a}\mathit{1}}^2+12{\mathit{a}\mathit{2}}^3}}+({\mathit{a}\mathit{1}}^2-3\mathit{a}\mathit{2})(i\sqrt{3}-1))\frac{1}{\sqrt[3]{36\mathit{a}\mathit{2}\mathit{a}\mathit{1}-108\mathit{a}\mathit{3}-8{\mathit{a}\mathit{1}}^3+12\sqrt{81{\mathit{a}\mathit{3}}^2+(12{\mathit{a}\mathit{1}}^3-54\mathit{a}\mathit{2}\mathit{a}\mathit{1})\mathit{a}\mathit{3}-3{\mathit{a}\mathit{2}}^2{\mathit{a}\mathit{1}}^2+12{\mathit{a}\mathit{2}}^3}}}}\}$$ Mathematica raw input

DSolve[a3*y[x] + a2*y'[x] + a1*y''[x] + y'''[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> E^(x*Root[a3 + a2*#1 + a1*#1^2 + #1^3 & , 1])*C[1] + E^(x*Root[a3 + a2
*#1 + a1*#1^2 + #1^3 & , 2])*C[2] + E^(x*Root[a3 + a2*#1 + a1*#1^2 + #1^3 & , 3]
)*C[3]}}

Maple raw input

dsolve(diff(diff(diff(y(x),x),x),x)+a1*diff(diff(y(x),x),x)+a2*diff(y(x),x)+a3*y(x) = 0, y(x),'implicit')

Maple raw output

y(x) = _C3*exp(1/6*((36*a2*a1-108*a3-8*a1^3+12*(81*a3^2+(12*a1^3-54*a1*a2)*a3-3*
a2^2*a1^2+12*a2^3)^(1/2))^(2/3)-2*a1*(36*a2*a1-108*a3-8*a1^3+12*(81*a3^2+(12*a1^
3-54*a1*a2)*a3-3*a2^2*a1^2+12*a2^3)^(1/2))^(1/3)+4*a1^2-12*a2)/(36*a2*a1-108*a3-
8*a1^3+12*(81*a3^2+(12*a1^3-54*a1*a2)*a3-3*a2^2*a1^2+12*a2^3)^(1/2))^(1/3)*x)+_C
2*exp(-1/3*((-1/4*I*3^(1/2)+1/4)*(36*a2*a1-108*a3-8*a1^3+12*(81*a3^2+(12*a1^3-54
*a1*a2)*a3-3*a2^2*a1^2+12*a2^3)^(1/2))^(2/3)+a1*(36*a2*a1-108*a3-8*a1^3+12*(81*a
3^2+(12*a1^3-54*a1*a2)*a3-3*a2^2*a1^2+12*a2^3)^(1/2))^(1/3)+(a1^2-3*a2)*(I*3^(1/
2)+1))*x/(36*a2*a1-108*a3-8*a1^3+12*(81*a3^2+(12*a1^3-54*a1*a2)*a3-3*a2^2*a1^2+1
2*a2^3)^(1/2))^(1/3))+_C1*exp(1/3*((-1/4*I*3^(1/2)-1/4)*(36*a2*a1-108*a3-8*a1^3+
12*(81*a3^2+(12*a1^3-54*a1*a2)*a3-3*a2^2*a1^2+12*a2^3)^(1/2))^(2/3)-a1*(36*a2*a1
-108*a3-8*a1^3+12*(81*a3^2+(12*a1^3-54*a1*a2)*a3-3*a2^2*a1^2+12*a2^3)^(1/2))^(1/
3)+(a1^2-3*a2)*(I*3^(1/2)-1))*x/(36*a2*a1-108*a3-8*a1^3+12*(81*a3^2+(12*a1^3-54*
a1*a2)*a3-3*a2^2*a1^2+12*a2^3)^(1/2))^(1/3))