Internal problem ID [10478]
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]]
\[ \boxed {x^{\prime \prime }+x^{\prime } t +x=0} \] 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: 0.016 (sec). Leaf size: 33
dsolve([diff(x(t),t$2)+t*diff(x(t),t)+x(t)=0,exp(-t^2/2)],x(t), singsol=all)
\[ x \left (t \right ) = \operatorname {erf}\left (\frac {i \sqrt {2}\, t}{2}\right ) {\mathrm e}^{-\frac {t^{2}}{2}} c_{1} +c_{2} {\mathrm e}^{-\frac {t^{2}}{2}} \]
✓ Solution by Mathematica
Time used: 0.019 (sec). Leaf size: 34
DSolve[x''[t]+t*x'[t]+x[t]==0,x[t],t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \sqrt {2} c_1 \operatorname {DawsonF}\left (\frac {t}{\sqrt {2}}\right )+c_2 e^{-\frac {t^2}{2}} \\ \end{align*}