82.27.4 problem Ex. 4

Internal problem ID [18808]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter VI. Linear equations with constant coefficients. problems at page 73
Problem number : Ex. 4
Date solved : Monday, March 31, 2025 at 06:15:27 PM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime }-y^{\prime \prime }-8 y^{\prime }+12 y&=X \left (x \right ) \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 64

ode:=diff(diff(diff(y(x),x),x),x)-diff(diff(y(x),x),x)-8*diff(y(x),x)+12*y(x) = X(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{-3 x} \left (\left (5 \int X \left (x \right ) {\mathrm e}^{-2 x}d x x +25 c_3 x -\int \left (5 x +1\right ) X \left (x \right ) {\mathrm e}^{-2 x}d x +25 c_2 \right ) {\mathrm e}^{5 x}+\int X \left (x \right ) {\mathrm e}^{3 x}d x +25 c_1 \right )}{25} \]

✓ Mathematica. Time used: 0.045 (sec). Leaf size: 110

ode=D[y[x],{x,3}]-D[y[x],{x,2}]-8*D[y[x],x]+12*y[x]==X[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to e^{-3 x} \left (\int _1^x\frac {1}{25} e^{3 K[1]} X(K[1])dK[1]+e^{5 x} \int _1^x-\frac {1}{25} e^{-2 K[2]} (5 K[2]+1) X(K[2])dK[2]+e^{5 x} x \int _1^x\frac {1}{5} e^{-2 K[3]} X(K[3])dK[3]+c_2 e^{5 x}+c_3 e^{5 x} x+c_1\right ) \]

✓ Sympy. Time used: 2.925 (sec). Leaf size: 60

from sympy import * 
x = symbols("x") 
y = Function("y") 
X = Function("X") 
ode = Eq(-X(x) + 12*y(x) - 8*Derivative(y(x), x) - Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 3)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (C_{1} + \frac {\int X{\left (x \right )} e^{3 x}\, dx}{25}\right ) e^{- 3 x} + \left (C_{2} + x \left (C_{3} + \frac {\int X{\left (x \right )} e^{- 2 x}\, dx}{5}\right ) - \frac {\int \left (5 x + 1\right ) X{\left (x \right )} e^{- 2 x}\, dx}{25}\right ) e^{2 x} \]