89.27.5 problem 5

Internal problem ID [24887]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 13. Systems of equations. Exercises at page 200
Problem number : 5
Date solved : Thursday, October 02, 2025 at 10:49:05 PM
CAS classification : system_of_ODEs

\begin{align*} 3 v^{\prime }\left (x \right )+2 v \left (x \right )+w^{\prime }\left (x \right )-6 w \left (x \right )&=5 \,{\mathrm e}^{x}\\ 4 v^{\prime }\left (x \right )+2 v \left (x \right )+w^{\prime }\left (x \right )-8 w \left (x \right )&=5 \,{\mathrm e}^{x}+2 x -3 \end{align*}

✓ Maple. Time used: 0.103 (sec). Leaf size: 71

ode:=[3*diff(v(x),x)+2*v(x)+diff(w(x),x)-6*w(x) = 5*exp(x), 4*diff(v(x),x)+2*v(x)+diff(w(x),x)-8*w(x) = 5*exp(x)+2*x-3]; 
dsolve(ode);
 

\begin{align*} v \left (x \right ) &= \sin \left (2 x \right ) c_2 +\cos \left (2 x \right ) c_1 -3 x +2 \,{\mathrm e}^{x}+5+5 \cos \left (2 x \right )+\frac {3 \sin \left (2 x \right )}{2} \\ w \left (x \right ) &= \cos \left (2 x \right ) c_2 -\sin \left (2 x \right ) c_1 +{\mathrm e}^{x}-5 \sin \left (2 x \right )+\frac {3 \cos \left (2 x \right )}{2}-x \\ \end{align*}

✓ Mathematica. Time used: 0.167 (sec). Leaf size: 54

ode={3*D[v[x],x]+2*v[x]+D[w[x],x]-6*w[x]==5*Exp[x],4*D[v[x],x]+2*v[x]+D[w[x],x]-8*w[x]==5*Exp[x]+2*x-3}; 
ic={}; 
DSolve[{ode,ic},{w[x],v[x]},x,IncludeSingularSolutions->True]
 

\begin{align*} v(x)&\to -3 x+2 e^x+c_1 \cos (2 x)+c_2 \sin (2 x)+5\\ w(x)&\to -x+e^x+c_2 \cos (2 x)-c_1 \sin (2 x) \end{align*}

✓ Sympy. Time used: 0.174 (sec). Leaf size: 129

from sympy import * 
x = symbols("x") 
w = Function("w") 
v = Function("v") 
ode=[Eq(2*v(x) - 6*w(x) - 5*exp(x) + 3*Derivative(v(x), x) + Derivative(w(x), x),0),Eq(-2*x + 2*v(x) - 8*w(x) - 5*exp(x) + 4*Derivative(v(x), x) + Derivative(w(x), x) + 3,0)] 
ics = {} 
dsolve(ode,func=[w(x),v(x)],ics=ics)
 

\[ \left [ w{\left (x \right )} = - C_{1} \sin {\left (2 x \right )} - C_{2} \cos {\left (2 x \right )} - x \sin ^{2}{\left (2 x \right )} - x \cos ^{2}{\left (2 x \right )} + e^{x} \sin ^{2}{\left (2 x \right )} + e^{x} \cos ^{2}{\left (2 x \right )}, \ v{\left (x \right )} = C_{1} \cos {\left (2 x \right )} - C_{2} \sin {\left (2 x \right )} - 3 x \sin ^{2}{\left (2 x \right )} - 3 x \cos ^{2}{\left (2 x \right )} + 2 e^{x} \sin ^{2}{\left (2 x \right )} + 2 e^{x} \cos ^{2}{\left (2 x \right )} + 5 \sin ^{2}{\left (2 x \right )} + 5 \cos ^{2}{\left (2 x \right )}\right ] \]