Internal problem ID [2283]
Book: Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition,
2015
Section: Chapter 8, Linear differential equations of order n. Section 8.7, The Variation of Parameters
Method. page 556
Problem number: Problem 19.
ODE order: 3.
ODE degree: 1.
CAS Maple gives this as type [[_3rd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime \prime }-3 y^{\prime \prime }+3 y^{\prime }-y-\frac {2 \,{\mathrm e}^{x}}{x^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 28
dsolve(diff(y(x),x$3)-3*diff(y(x),x$2)+3*diff(y(x),x)-y(x)=2*x^(-2)*exp(x),y(x), singsol=all)
\[ y \relax (x ) = -2 \,{\mathrm e}^{x} \ln \relax (x ) x +{\mathrm e}^{x} c_{1}+c_{2} {\mathrm e}^{x} x +c_{3} x^{2} {\mathrm e}^{x} \]
✓ Solution by Mathematica
Time used: 0.382 (sec). Leaf size: 627
DSolve[y'''[x]-6*y''[x]+3*y'[x]-y[x]==2*x^(-2)*Exp[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {2 i \left (\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+3 \text {$\#$1}-1\&,1\right ]-\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+3 \text {$\#$1}-1\&,2\right ]\right ) \exp \left (x \text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+3 \text {$\#$1}-1\&,3\right ]\right ) \left (-\frac {\exp \left (x-x \text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+3 \text {$\#$1}-1\&,3\right ]\right )}{x}-\left (\left (-1+\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+3 \text {$\#$1}-1\&,3\right ]\right ) \text {Ei}\left (x-x \text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+3 \text {$\#$1}-1\&,3\right ]\right )\right )\right )}{3 \sqrt {39}}+\frac {2 i \left (\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+3 \text {$\#$1}-1\&,2\right ]-\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+3 \text {$\#$1}-1\&,3\right ]\right ) \exp \left (x \text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+3 \text {$\#$1}-1\&,1\right ]\right ) \left (\left (\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+3 \text {$\#$1}-1\&,2\right ]+\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+3 \text {$\#$1}-1\&,3\right ]-5\right ) \text {Ei}\left (x \left (-5+\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+3 \text {$\#$1}-1\&,2\right ]+\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+3 \text {$\#$1}-1\&,3\right ]\right )\right )-\frac {\exp \left (x \left (\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+3 \text {$\#$1}-1\&,2\right ]+\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+3 \text {$\#$1}-1\&,3\right ]-5\right )\right )}{x}\right )}{3 \sqrt {39}}-\frac {2 i \left (\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+3 \text {$\#$1}-1\&,1\right ]-\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+3 \text {$\#$1}-1\&,3\right ]\right ) \exp \left (x \text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+3 \text {$\#$1}-1\&,2\right ]\right ) \left (-\frac {\exp \left (x-x \text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+3 \text {$\#$1}-1\&,2\right ]\right )}{x}-\left (\left (-1+\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+3 \text {$\#$1}-1\&,2\right ]\right ) \text {Ei}\left (x-x \text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+3 \text {$\#$1}-1\&,2\right ]\right )\right )\right )}{3 \sqrt {39}}+c_2 \exp \left (x \text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+3 \text {$\#$1}-1\&,2\right ]\right )+c_3 \exp \left (x \text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+3 \text {$\#$1}-1\&,3\right ]\right )+c_1 \exp \left (x \text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+3 \text {$\#$1}-1\&,1\right ]\right ) \\ \end{align*}