64.22.7 problem 3(c)

Internal problem ID [13672]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 11, The nth order homogeneous linear differential equation. Section 11.8, Exercises page 583
Problem number : 3(c)
Date solved : Tuesday, January 28, 2025 at 05:54:46 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \frac {\left (1+t \right ) x^{\prime \prime }}{t}-\frac {x^{\prime }}{t^{2}}+\frac {x}{t^{3}}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 19

dsolve((t+1)/t*diff(x(t),t$2)-1/t^2*diff(x(t),t)+1/t^3*x(t)=0,x(t), singsol=all)
 

\[ x \left (t \right ) = t \left (c_{2} \ln \left (t +1\right )-c_{2} \ln \left (t \right )+c_{1} \right ) \]

✓ Solution by Mathematica

Time used: 0.163 (sec). Leaf size: 103

DSolve[(t+1)/t*D[x[t],{t,2}]-1/t^2*D[x[t],t]+1/t^3*x[t]==0,{x[t]},t,IncludeSingularSolutions -> True]
 

\[ x(t)\to \exp \left (\int _1^t\frac {2 K[1]+1}{2 K[1]^2+2 K[1]}dK[1]-\frac {1}{2} \int _1^t-\frac {1}{K[2]^2+K[2]}dK[2]\right ) \left (c_2 \int _1^t\exp \left (-2 \int _1^{K[3]}\frac {2 K[1]+1}{2 K[1]^2+2 K[1]}dK[1]\right )dK[3]+c_1\right ) \]