Internal problem ID [10481]
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: 5.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{\prime \prime }-\frac {\left (t +2\right ) x^{\prime }}{t}+\frac {\left (t +2\right ) x}{t^{2}}=0} \] 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: 13
dsolve([diff(x(t),t$2)-(t+2)/t*diff(x(t),t)+(t+2)/t^2*x(t)=0,t],x(t), singsol=all)
\[ x \left (t \right ) = c_{1} t +c_{2} {\mathrm e}^{t} t \]
✓ Solution by Mathematica
Time used: 0.007 (sec). Leaf size: 16
DSolve[x''[t]-(t+2)/t*x'[t]+(t+2)/t^2*x[t]==0,x[t],t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to t \left (c_2 e^t+c_1\right ) \\ \end{align*}