Internal problem ID [11497]
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]
\[ \boxed {x^{\prime \prime }-t x^{\prime }+x=0} \] 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],singsol=all)
\[ x \left (t \right ) = c_{2} {\mathrm e}^{\frac {t^{2}}{2}}+\frac {\left (i c_{2} \sqrt {2}\, \sqrt {\pi }\, \operatorname {erf}\left (\frac {i \sqrt {2}\, t}{2}\right )+2 c_{1} \right ) t}{2} \]
✓ Solution by Mathematica
Time used: 0.09 (sec). Leaf size: 61
DSolve[x''[t]-t*x'[t]+x[t]==0,x[t],t,IncludeSingularSolutions -> True]
\[ x(t)\to -\sqrt {\frac {\pi }{2}} c_2 \sqrt {t^2} \text {erfi}\left (\frac {\sqrt {t^2}}{\sqrt {2}}\right )+c_2 e^{\frac {t^2}{2}}+\sqrt {2} c_1 t \]