63.13.2 problem 2

Internal problem ID [13173]
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
Date solved : Tuesday, January 28, 2025 at 05:11:58 AM
CAS classification : [_Hermite]

\begin{align*} x^{\prime \prime }-t x^{\prime }+x&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} x&=t \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 37

dsolve([diff(x(t),t$2)-t*diff(x(t),t)+x(t)=0,t],singsol=all)
 

\[ x \left (t \right ) = i \sqrt {2}\, \sqrt {\pi }\, {\mathrm e}^{\frac {t^{2}}{2}} c_{2} -t \left (\operatorname {erf}\left (\frac {i \sqrt {2}\, t}{2}\right ) \pi c_{2} -c_{1} \right ) \]

✓ Solution by Mathematica

Time used: 0.049 (sec). Leaf size: 61

DSolve[D[x[t],{t,2}]-t*D[x[t],t]+x[t]==0,x[t],t,IncludeSingularSolutions -> True]
 

\[ x(t)\to -\sqrt {\frac {\pi }{2}} c_2 \sqrt {t^2} \text {erfi}\left (\frac {\sqrt {t^2}}{\sqrt {2}}\right )+c_2 e^{\frac {t^2}{2}}+\sqrt {2} c_1 t \]