71.18.6 problem 5 c
Internal
problem
ID
[14501]
Book
:
Ordinary
Differential
Equations
by
Charles
E.
Roberts,
Jr.
CRC
Press.
2010
Section
:
Chapter
8.
Linear
Systems
of
First-Order
Differential
Equations.
Exercises
8.3
page
379
Problem
number
:
5
c
Date
solved
:
Thursday, March 13, 2025 at 03:31:44 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d x}y_{1} \left (x \right )&=2 y_{1} \left (x \right )-3 y_{2} \left (x \right )+4 x -2\\ \frac {d}{d x}y_{2} \left (x \right )&=y_{1} \left (x \right )-2 y_{2} \left (x \right )+3 x \end{align*}
✓ Maple. Time used: 0.039 (sec). Leaf size: 35
ode:=[diff(y__1(x),x) = 2*y__1(x)-3*y__2(x)+4*x-2, diff(y__2(x),x) = y__1(x)-2*y__2(x)+3*x];
dsolve(ode);
\begin{align*}
y_{1} \left (x \right ) &= c_{2} {\mathrm e}^{-x}+{\mathrm e}^{x} c_{1} +x \\
y_{2} \left (x \right ) &= c_{2} {\mathrm e}^{-x}+\frac {{\mathrm e}^{x} c_{1}}{3}-1+2 x \\
\end{align*}
✓ Mathematica. Time used: 1.445 (sec). Leaf size: 333
ode={D[ y1[x],x]==-2*y1[x]-3*y2[x]+4*x-2,D[ y2[x],x]==y1[x]-2*y2[x]+3*x};
ic={};
DSolve[{ode,ic},{y1[x],y2[x]},x,IncludeSingularSolutions->True]
\begin{align*}
\text {y1}(x)\to e^{-2 x} \left (\cos \left (\sqrt {3} x\right ) \int _1^xe^{2 K[1]} \left (\cos \left (\sqrt {3} K[1]\right ) (4 K[1]-2)+3 \sqrt {3} K[1] \sin \left (\sqrt {3} K[1]\right )\right )dK[1]-\sqrt {3} \sin \left (\sqrt {3} x\right ) \int _1^x\frac {1}{3} e^{2 K[2]} \left (9 \cos \left (\sqrt {3} K[2]\right ) K[2]+2 \sqrt {3} (1-2 K[2]) \sin \left (\sqrt {3} K[2]\right )\right )dK[2]+c_1 \cos \left (\sqrt {3} x\right )-\sqrt {3} c_2 \sin \left (\sqrt {3} x\right )\right ) \\
\text {y2}(x)\to \frac {1}{3} e^{-2 x} \left (3 \cos \left (\sqrt {3} x\right ) \int _1^x\frac {1}{3} e^{2 K[2]} \left (9 \cos \left (\sqrt {3} K[2]\right ) K[2]+2 \sqrt {3} (1-2 K[2]) \sin \left (\sqrt {3} K[2]\right )\right )dK[2]+\sqrt {3} \sin \left (\sqrt {3} x\right ) \int _1^xe^{2 K[1]} \left (\cos \left (\sqrt {3} K[1]\right ) (4 K[1]-2)+3 \sqrt {3} K[1] \sin \left (\sqrt {3} K[1]\right )\right )dK[1]+3 c_2 \cos \left (\sqrt {3} x\right )+\sqrt {3} c_1 \sin \left (\sqrt {3} x\right )\right ) \\
\end{align*}
✓ Sympy. Time used: 0.178 (sec). Leaf size: 32
from sympy import *
x = symbols("x")
y__1 = Function("y__1")
y__2 = Function("y__2")
ode=[Eq(-4*x - 2*y__1(x) + 3*y__2(x) + Derivative(y__1(x), x) + 2,0),Eq(-3*x - y__1(x) + 2*y__2(x) + Derivative(y__2(x), x),0)]
ics = {}
dsolve(ode,func=[y__1(x),y__2(x)],ics=ics)
\[
\left [ y^{1}{\left (x \right )} = C_{1} e^{- x} + 3 C_{2} e^{x} + x, \ y^{2}{\left (x \right )} = C_{1} e^{- x} + C_{2} e^{x} + 2 x - 1\right ]
\]