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Example: Solving Vector Differential Equations

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Wize University Linear Algebra Textbook > Differential Equations

Systems of Linear Differential Equations

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A set of differential equations is given by:
dXdt=X(t)−4Y(t)\frac{dX}{dt}=X\left(t\right)-4Y\left(t\right)

dYdt=−2X(t)+3Y(t)\frac{dY}{dt}=-2X\left(t\right)+3Y\left(t\right)
a) Express these differential equation in the matrix formdV⃗dt=AV⃗\frac{d\vec V}{dt}=A\vec V
b) Find eigenvalues and eigenvectors of the matrix A
c) Express the general solution to this system
d) Assume at t=0, both X and Y are equal to 1. Find X(t) and Y(t).













More Systems of Linear Differential Equations Questions:
Practice: Solving a System of Linear DEs
Solve the following linear system of differential equations:
{y1′=2y1−3y2y2′=−6y1−y2\left\{\begin{array}{rcrcr} y_1'&=&2y_1&-&3y_2\\ y_2'&=&-6y_1&-&y_2\\ \end{array}\right.
Practice: Solving a System of Linear DEs (Initial Value Problem)
Find the general solution of the following linear system of differential equations:
{y1′=−5y1+y2y2′=8y1+2y2\left\{\begin{array}{rcrcr} y_1'&=&-5y_1&+&y_2\\ y_2'&=&8y_1&+&2y_2\\ \end{array}\right.
Practice: Higher Order DEs
Write the following differential equation as a linear system of differential equations. You do not need to solve the system.
y′′′+3y′′−8y′+2y=0y'''+3y''-8y'+2y=0
Second order linear differential equation
Write the following equation as an equivalent system of linear differential equations:
y′′′−3y′′+2y′+8y=0y'''-3y''+2y'+8y=0
263_final_#2_18.1W_eg31
Solve the system of DEs given by x⃗=(10021−2321)⋅x⃗\vec{x} = \begin{pmatrix} 1 & 0 & 0 \\ 2 & 1 & -2 \\ 3 & 2 & 1\end{pmatrix} \cdot \vec{x}.
263_final_#2_18.1W_eg30
Solve the system of DEs given by{x1′=x1−x2x2′=5x1−3x2\begin{cases} x_1' & = x_1 - x_2 \\ x_2' & = 5x_1 -3x_2 \end{cases}.
263_final_#2_18.1W_eg29
Solve the system of DEs given by x⃗=(−1−41−1)⋅x⃗\vec{x} = \begin{pmatrix} -1 & -4 \\ 1 & -1\end{pmatrix} \cdot \vec{x}.
263_final_#2_18.1W_eg27
Solve the system of DEs given by x⃗=(11121−1−8−5−3)⋅x⃗\vec{x} = \begin{pmatrix} 1 & 1 & 1\\ 2 & 1 & -1 \\ -8 & -5 & -3\end{pmatrix} \cdot \vec{x}.
263_final_#2_18.1W_eg26
Solve the system of DEs given by x⃗=(3−22−2)⋅x⃗\vec{x} = \begin{pmatrix} 3 & -2 \\ 2 & -2\end{pmatrix} \cdot \vec{x}.
263_final_#2_18.1W_eg28
Solve the system of DEs given byx⃗=(2−51−2)⋅x⃗\vec{x} = \begin{pmatrix} 2 & -5 \\ 1 & - 2\end{pmatrix} \cdot \vec{x}.
263_final_#2_18.1W_eg25
Solve the system of DEs given by x⃗=(114−2)⋅x⃗\vec{x} = \begin{pmatrix} 1 & 1 \\ 4 & -2\end{pmatrix} \cdot \vec{x}.