64.11.40 problem 40

Internal problem ID [13419]
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 : 40
Date solved : Monday, March 31, 2025 at 07:53:46 AM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime }-6 y^{\prime \prime }+9 y^{\prime }-4 y&=8 x^{2}+3-6 \,{\mathrm e}^{2 x} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=7\\ y^{\prime \prime }\left (0\right )&=0 \end{align*}

✓ Maple. Time used: 0.029 (sec). Leaf size: 35

ode:=diff(diff(diff(y(x),x),x),x)-6*diff(diff(y(x),x),x)+9*diff(y(x),x)-4*y(x) = 8*x^2+3-6*exp(2*x); 
ic:=y(0) = 1, D(y)(0) = 7, (D@@2)(y)(0) = 0; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = -2 x^{2}-9 x +3 \,{\mathrm e}^{2 x}-15+\frac {44 \,{\mathrm e}^{x}}{3}-\frac {5 \,{\mathrm e}^{4 x}}{3}+2 \,{\mathrm e}^{x} x \]

✓ Mathematica. Time used: 0.284 (sec). Leaf size: 258

ode=D[y[x],{x,3}]-6*D[y[x],{x,2}]+9*D[y[x],x]-4*y[x]==8*x^2+3-6*Exp[2*x]; 
ic={y[0]==1,Derivative[1][y][0] ==7,Derivative[2][y][0] ==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{9} e^x \left (-9 x \int _1^0\frac {1}{3} e^{-K[2]} \left (-8 K[2]^2+6 e^{2 K[2]}-3\right )dK[2]+9 x \int _1^x\frac {1}{3} e^{-K[2]} \left (-8 K[2]^2+6 e^{2 K[2]}-3\right )dK[2]+9 \int _1^x\frac {1}{9} e^{-K[1]} (3 K[1]-1) \left (8 K[1]^2-6 e^{2 K[1]}+3\right )dK[1]-9 e^{3 x} \int _1^0\frac {1}{9} e^{-4 K[3]} \left (8 K[3]^2-6 e^{2 K[3]}+3\right )dK[3]+9 e^{3 x} \int _1^x\frac {1}{9} e^{-4 K[3]} \left (8 K[3]^2-6 e^{2 K[3]}+3\right )dK[3]-9 \int _1^0\frac {1}{9} e^{-K[1]} (3 K[1]-1) \left (8 K[1]^2-6 e^{2 K[1]}+3\right )dK[1]+93 x-13 e^{3 x}+22\right ) \]

✓ Sympy. Time used: 0.288 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-8*x**2 - 4*y(x) + 6*exp(2*x) + 9*Derivative(y(x), x) - 6*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 3)) - 3,0) 
ics = {y(0): 1, Subs(Derivative(y(x), x), x, 0): 7, Subs(Derivative(y(x), (x, 2)), x, 0): 0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = - 2 x^{2} - 9 x + \left (2 x + \frac {44}{3}\right ) e^{x} - \frac {5 e^{4 x}}{3} + 3 e^{2 x} - 15 \]