64.11.46 problem 46

Internal problem ID [13417]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 4, Section 4.3. The method of undetermined coefficients. Exercises page 151
Problem number : 46
Date solved : Wednesday, March 05, 2025 at 09:56:40 PM
CAS classification : [[_3rd_order, _missing_y]]

\begin{align*} y^{\prime \prime \prime }-3 y^{\prime \prime }+2 y^{\prime }&=x^{2} {\mathrm e}^{x}+3 x \,{\mathrm e}^{2 x}+5 x^{2} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 55

ode:=diff(diff(diff(y(x),x),x),x)-3*diff(diff(y(x),x),x)+2*diff(y(x),x) = x^2*exp(x)+3*exp(2*x)*x+5*x^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (6 x^{2}+4 c_{1} -18 x +21\right ) {\mathrm e}^{2 x}}{8}+\frac {\left (-x^{3}+3 c_{2} -6 x +6\right ) {\mathrm e}^{x}}{3}+\frac {5 x^{3}}{6}+\frac {15 x^{2}}{4}+\frac {35 x}{4}+c_{3} \]

✓ Mathematica. Time used: 21.78 (sec). Leaf size: 98

ode=D[y[x],{x,3}]-3*D[y[x],{x,2}]+2*D[y[x],x]==x^2*Exp[x]+3*x*Exp[2*x]+5*x^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \int _1^xe^{K[3]} \left (c_1+e^{K[3]} c_2+\int _1^{K[3]}\left (-e^{-K[1]} \left (5+e^{K[1]}\right ) K[1]^2-3 e^{K[1]} K[1]\right )dK[1]+e^{K[3]} \int _1^{K[3]}K[2] \left (e^{-2 K[2]} \left (5+e^{K[2]}\right ) K[2]+3\right )dK[2]\right )dK[3]+c_3 \]

✓ Sympy. Time used: 0.393 (sec). Leaf size: 54

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**2*exp(x) - 5*x**2 - 3*x*exp(2*x) + 2*Derivative(y(x), x) - 3*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 3)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} + \frac {5 x^{3}}{6} + \frac {15 x^{2}}{4} + \frac {35 x}{4} + \left (C_{2} + \frac {3 x^{2}}{4} - \frac {9 x}{4}\right ) e^{2 x} + \left (C_{3} - \frac {x^{3}}{3} - 2 x\right ) e^{x} \]