Internal problem ID [10479]
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: 2.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Hermite]
\[ \boxed {x^{\prime \prime }-x^{\prime } t +x=0} \] Given that one solution of the ode is \begin {align*} x_1 &= t \end {align*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 38
dsolve([diff(x(t),t$2)-t*diff(x(t),t)+x(t)=0,t],x(t), singsol=all)
\[ x \left (t \right ) = c_{1} t +c_{2} \left (i \sqrt {2}\, \sqrt {\pi }\, {\mathrm e}^{\frac {t^{2}}{2}}-\pi \,\operatorname {erf}\left (\frac {i \sqrt {2}\, t}{2}\right ) t \right ) \]
✓ Solution by Mathematica
Time used: 0.006 (sec). Leaf size: 47
DSolve[x''[t]-t*x'[t]+x[t]==0,x[t],t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {t \left (2 c_1-\sqrt {\pi } c_2 \text {erfi}\left (\frac {t}{\sqrt {2}}\right )\right )}{\sqrt {2}}+c_2 e^{\frac {t^2}{2}} \\ \end{align*}