Internal problem ID [11496]
Book: A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag,
NY. 2015.
Section: Chapter 2, Second order linear equations. Section 2.4.3 Reduction of order. Exercises
page 125
Problem number: 1.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _exact, _linear, _homogeneous]]
\[ \boxed {x^{\prime \prime }+t x^{\prime }+x=0} \] Given that one solution of the ode is \begin {align*} x_1 &= {\mathrm e}^{-\frac {t^{2}}{2}} \end {align*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 23
dsolve([diff(x(t),t$2)+t*diff(x(t),t)+x(t)=0,exp(-t^2/2)],singsol=all)
\[ x \left (t \right ) = \left (\operatorname {erf}\left (\frac {i \sqrt {2}\, t}{2}\right ) c_{1} +c_{2} \right ) {\mathrm e}^{-\frac {t^{2}}{2}} \]
✓ Solution by Mathematica
Time used: 0.037 (sec). Leaf size: 41
DSolve[x''[t]+t*x'[t]+x[t]==0,x[t],t,IncludeSingularSolutions -> True]
\[ x(t)\to \frac {1}{2} e^{-\frac {t^2}{2}} \left (\sqrt {2 \pi } c_1 \text {erfi}\left (\frac {t}{\sqrt {2}}\right )+2 c_2\right ) \]