Question 7.3.3: Find a general solution of the system dx1/dt = 4x1 - 3x2, dx...
Find a general solution of the system
\frac{dx_{1}}{dt} = 4x_{1} – 3x_{2},
\frac{dx_{2}}{dt} = 3x_{1} + 4x_{2}. (21)
The coefficient matrix
A = \begin{bmatrix} 4 & -3 \\ 3 & 4 \end{bmatrix}
has the characteristic equation
|A – λI| = \begin{vmatrix} 4-\lambda & -3 \\ 3 & 4-\lambda \end{vmatrix} = (4 – λ)² + 9 =0
and hence has the complex conjugate eigenvalues λ = 4 – 3i and \bar{\lambda } = 4 + 3i . Substituting λ = 4 – 3i in the eigenvector equation (A – λI)v = 0, we get the equation
\left [ \begin{matrix} \textbf{A} – (4 – 3i) · \textbf{I} \end{matrix} \right ] \textbf{v} = \begin{bmatrix} 3i & -3 \\ 3 & 3i \end{bmatrix} \left [ \begin{matrix} a \\ b \end{matrix} \right ] = \left [ \begin{matrix} 0 \\ 0 \end{matrix} \right ]
for an associated eigenvalue v = [a b ]^{T} . Division of each row by 3 yields the two scalar equations
ia – b = 0,
a + ib = 0,
each of which is satisfied by a = 1 and b = i . Thus, v = [1 i]^{T} is a complex eigenvector associated with the complex eigenvalue λ = 4 – 3i . The corresponding complex-valued solution x(t) = ve^{λ_{t}} of x^{′ } = Ax is then
x(t) = \left [ \begin{matrix} 1 \\ i \end{matrix} \right ] e^{(4-3i)t} =\left [ \begin{matrix} 1 \\ i \end{matrix} \right ] e^{4t} (\cos 3t – i \sin 3t) = e^{4t} \left [ \begin{matrix} \cos 3t – i \sin 3t \\ i \cos 3t + \sin 3t \end{matrix} \right ] .
The real and imaginary parts of x(t) are the real-valued solutions
x_{1}(t) = e^{4t} \left [ \begin{matrix}\cos 3t \\ \sin 3t \end{matrix} \right ] and x_{2}(t) = e^{4t} \left [ \begin{matrix} -\sin 3t \\ \cos 3t \end{matrix} \right ]
A real-valued general solution of x^{′} = Ax is then given by
\textbf{x}(t) = c_{1} \textbf{x}_{1}(t) + c_{2} \textbf{x}_{2}(t) = e^{4t} \left [ \begin{matrix} c_{1} \cos 3t – c_{2} \sin 3t \\ c_{1} \sin 3t + c_{2} \cos 3t \end{matrix} \right ] .Finally, a general solution of the system in (21) in scalar form is
x_{1}(t) = e^{4t} (c_{1} \cos 3t – c_{2} \sin 3t ),
x_{2}(t) = e^{4t} (c_{1} \sin 3t + c_{2} \cos 3t ).
Figure 7.3.4 shows some typical solution curves of the system in (21). Each appears to spiral counterclockwise as it emanates from the origin in the x_{1}x_{2}-plane. Actually, because of the factor e^{4t} in the general solution, we see that
- Along each solution curve, the point (x_{1}(t), x_{2}(t)) approaches the origin as t → – \infty,whereas
- The absolute values of x_{1}(t) and x_{2}(t) both increase without bound as t → + \infty.
