Internal problem ID [12931]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 2. Integration and differential equations. Additional exercises. page
32
Problem number: 2.2 (i).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }+3 y^{\prime }+8 y={\mathrm e}^{-x^{2}}} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 121
dsolve(diff(y(x),x$2)+3*diff(y(x),x)+8*y(x)=exp(-x^2),y(x), singsol=all)
\[ y = \sin \left (\frac {\sqrt {23}\, x}{2}\right ) {\mathrm e}^{-\frac {3 x}{2}} c_{2} +{\mathrm e}^{-\frac {3 x}{2}} \cos \left (\frac {\sqrt {23}\, x}{2}\right ) c_{1} +\frac {\sqrt {23}\, {\mathrm e}^{-\frac {3 x}{2}-\frac {7}{8}} \sqrt {\pi }\, \left ({\mathrm e}^{\frac {3 i \sqrt {23}}{8}} \left (i \cos \left (\frac {\sqrt {23}\, x}{2}\right )+\sin \left (\frac {\sqrt {23}\, x}{2}\right )\right ) \operatorname {erf}\left (x -\frac {3}{4}-\frac {i \sqrt {23}}{4}\right )-{\mathrm e}^{-\frac {3 i \sqrt {23}}{8}} \left (i \cos \left (\frac {\sqrt {23}\, x}{2}\right )-\sin \left (\frac {\sqrt {23}\, x}{2}\right )\right ) \operatorname {erf}\left (x -\frac {3}{4}+\frac {i \sqrt {23}}{4}\right )\right )}{46} \]
✓ Solution by Mathematica
Time used: 0.519 (sec). Leaf size: 205
DSolve[y''[x]+3*y'[x]+8*y[x]==Exp[-x^2],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{46} e^{\frac {1}{8} \left (-12 x-3 i \sqrt {23}-7\right )} \left (-i e^{\frac {3 i \sqrt {23}}{4}} \sqrt {23 \pi } \text {erf}\left (-x+\frac {i \sqrt {23}}{4}+\frac {3}{4}\right ) \left (\cos \left (\frac {\sqrt {23} x}{2}\right )-i \sin \left (\frac {\sqrt {23} x}{2}\right )\right )+i \sqrt {23 \pi } \text {erf}\left (\frac {1}{4} \left (-4 x-i \sqrt {23}+3\right )\right ) \left (\cos \left (\frac {\sqrt {23} x}{2}\right )+i \sin \left (\frac {\sqrt {23} x}{2}\right )\right )+46 e^{\frac {7}{8}+\frac {3 i \sqrt {23}}{8}} \left (c_2 \cos \left (\frac {\sqrt {23} x}{2}\right )+c_1 \sin \left (\frac {\sqrt {23} x}{2}\right )\right )\right ) \]