Internal problem ID [10480]
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: 4.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x]]
\[ \boxed {x^{\prime \prime }-2 a x^{\prime }+a^{2} x=0} \] Given that one solution of the ode is \begin {align*} x_1 &= {\mathrm e}^{a t} \end {align*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 18
dsolve([diff(x(t),t$2)-2*a*diff(x(t),t)+a^2*x(t)=0,exp(a*t)],x(t), singsol=all)
\[ x \left (t \right ) = {\mathrm e}^{a t} c_{1} +c_{2} {\mathrm e}^{a t} t \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 18
DSolve[x''[t]-2*a*x'[t]+a^2*x[t]==0,x[t],t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to e^{a t} (c_2 t+c_1) \\ \end{align*}