Internal problem ID [11731]
Book: AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C.
ROBINSON. Cambridge University Press 2004
Section: Chapter 17, Reduction of order. Exercises page 162
Problem number: 17.4.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\left (-t^{2}+t \right ) x^{\prime \prime }+\left (-t^{2}+2\right ) x^{\prime }+\left (2-t \right ) x=0} \] Given that one solution of the ode is \begin {align*} x_1 &= {\mathrm e}^{-t} \end {align*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 16
dsolve([(t-t^2)*diff(x(t),t$2)+(2-t^2)*diff(x(t),t)+(2-t)*x(t)=0,exp(-t)],x(t), singsol=all)
\[ x \left (t \right ) = \frac {c_{1}}{t}+c_{2} {\mathrm e}^{-t} \]
✓ Solution by Mathematica
Time used: 0.104 (sec). Leaf size: 42
DSolve[(t-t^2)*x''[t]+(2-t^2)*x'[t]+(2-t)*x[t]==0,x[t],t,IncludeSingularSolutions -> True]
\[ x(t)\to \frac {e^{-t} \sqrt {1-t} \left (c_1 e^t-c_2 t\right )}{\sqrt {t-1} t} \]