64.21.3 problem 5

Internal problem ID [13664]
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 11.6, Exercises page 567
Problem number : 5
Date solved : Tuesday, January 28, 2025 at 05:54:33 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (t^{3}-2 t^{2}\right ) x^{\prime \prime }-\left (t^{3}+2 t^{2}-6 t \right ) x^{\prime }+\left (3 t^{2}-6\right ) x&=0 \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 15

dsolve((t^3-2*t^2)*diff(x(t),t$2)-(t^3+2*t^2-6*t)*diff(x(t),t)+(3*t^2-6)*x(t)=0,x(t), singsol=all)
 

\[ x \left (t \right ) = c_{1} t^{3}+c_{2} {\mathrm e}^{t} t \]

✓ Solution by Mathematica

Time used: 0.328 (sec). Leaf size: 115

DSolve[(t^3-2*t^2)*D[x[t],{t,2}]-(t^3+2*t^2-6*t)*D[x[t],t]+(3*t^2-6)*x[t]==0,{x[t]},t,IncludeSingularSolutions -> True]
 

\[ x(t)\to \exp \left (\int _1^t\frac {(K[1]-4) K[1]+2}{2 (K[1]-2) K[1]}dK[1]-\frac {1}{2} \int _1^t\left (-\frac {3}{K[2]}-1+\frac {1}{2-K[2]}\right )dK[2]\right ) \left (c_2 \int _1^t\exp \left (-2 \int _1^{K[3]}\frac {K[1]^2-4 K[1]+2}{2 (K[1]-2) K[1]}dK[1]\right )dK[3]+c_1\right ) \]