89.27.6 problem 14
Internal
problem
ID
[24888]
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
:
14
Date
solved
:
Thursday, October 02, 2025 at 10:49:06 PM
CAS
classification
:
system_of_ODEs
\begin{align*} 2 y^{\prime }+2 y+w^{\prime }\left (x \right )-w \left (x \right )&=1+x\\ y^{\prime }+3 y+w^{\prime }\left (x \right )+w \left (x \right )&=4 x +14 \end{align*}
✓ Maple. Time used: 0.061 (sec). Leaf size: 77
ode:=[2*diff(y(x),x)+2*y(x)+diff(w(x),x)-w(x) = 1+x, diff(y(x),x)+3*y(x)+diff(w(x),x)+w(x) = 4*x+14];
dsolve(ode);
\begin{align*}
w \left (x \right ) &= {\mathrm e}^{-x} \sin \left (2 x \right ) c_2 +{\mathrm e}^{-x} \cos \left (2 x \right ) c_1 +x +6 \\
y \left (x \right ) &= -\frac {{\mathrm e}^{-x} \sin \left (2 x \right ) c_2}{2}-\frac {{\mathrm e}^{-x} \cos \left (2 x \right ) c_2}{2}-\frac {{\mathrm e}^{-x} \cos \left (2 x \right ) c_1}{2}+\frac {{\mathrm e}^{-x} \sin \left (2 x \right ) c_1}{2}+2+x \\
\end{align*}
✓ Mathematica. Time used: 0.915 (sec). Leaf size: 136
ode={2*D[y[x],x]+2*y[x]+D[w[x],x]-w[x]==x+1,D[y[x],x]+2*y[x]+D[w[x],x]+w[x]==4*x+14};
ic={};
DSolve[{ode,ic},{w[x],y[x]},x,IncludeSingularSolutions->True]
\begin{align*} w(x)&\to \frac {3}{8} (4 x+19)+c_1 e^{-3 x/2} \cos \left (\frac {\sqrt {7} x}{2}\right )-\frac {(3 c_1+4 c_2) e^{-3 x/2} \sin \left (\frac {\sqrt {7} x}{2}\right )}{\sqrt {7}}\\ y(x)&\to \frac {5 x}{4}+c_2 e^{-3 x/2} \cos \left (\frac {\sqrt {7} x}{2}\right )+\frac {(4 c_1+3 c_2) e^{-3 x/2} \sin \left (\frac {\sqrt {7} x}{2}\right )}{\sqrt {7}}+\frac {33}{16} \end{align*}
✓ Sympy. Time used: 0.460 (sec). Leaf size: 231
from sympy import *
x = symbols("x")
w = Function("w")
y = Function("y")
ode=[Eq(-x - w(x) + 2*y(x) + Derivative(w(x), x) + 2*Derivative(y(x), x) - 1,0),Eq(-4*x + w(x) + 2*y(x) + Derivative(w(x), x) + Derivative(y(x), x) - 14,0)]
ics = {}
dsolve(ode,func=[w(x),y(x)],ics=ics)
\[
\left [ w{\left (x \right )} = \frac {3 x \sin ^{2}{\left (\frac {\sqrt {7} x}{2} \right )}}{2} + \frac {3 x \cos ^{2}{\left (\frac {\sqrt {7} x}{2} \right )}}{2} - \left (\frac {3 C_{1}}{4} + \frac {\sqrt {7} C_{2}}{4}\right ) e^{- \frac {3 x}{2}} \cos {\left (\frac {\sqrt {7} x}{2} \right )} - \left (\frac {\sqrt {7} C_{1}}{4} - \frac {3 C_{2}}{4}\right ) e^{- \frac {3 x}{2}} \sin {\left (\frac {\sqrt {7} x}{2} \right )} + \frac {57 \sin ^{2}{\left (\frac {\sqrt {7} x}{2} \right )}}{8} + \frac {57 \cos ^{2}{\left (\frac {\sqrt {7} x}{2} \right )}}{8}, \ y{\left (x \right )} = C_{1} e^{- \frac {3 x}{2}} \cos {\left (\frac {\sqrt {7} x}{2} \right )} - C_{2} e^{- \frac {3 x}{2}} \sin {\left (\frac {\sqrt {7} x}{2} \right )} + \frac {5 x \sin ^{2}{\left (\frac {\sqrt {7} x}{2} \right )}}{4} + \frac {5 x \cos ^{2}{\left (\frac {\sqrt {7} x}{2} \right )}}{4} + \frac {33 \sin ^{2}{\left (\frac {\sqrt {7} x}{2} \right )}}{16} + \frac {33 \cos ^{2}{\left (\frac {\sqrt {7} x}{2} \right )}}{16}\right ]
\]