Section 5.1. Introduction of system of differential equations

Objective:

1. Definition of a system of differential equations

2. Basic terms of a system differential equations

3. Converting a system of equation into one ODE and vice versa.

Definition: (a) A system of first order ordinary differential equations has the general form

$$\begin{array}{rl}{x}_1^{\prime }& =F_1(t,x_1,x_2,\dots ,x_n)\\ x_{2\prime }& =F_2(t,x_1,x_2,\dots ,x_n)\\ ⋮& =⋮\\ x_n^{\prime }& =F_2(t,x_1,x_2,\dots ,x_n)\end{array}$$

 where each $x_i$ is a function of $t$. 

(b) If each $F_i$ is a linear function of $x_1,...,x_n$, then the system of equations is said to be linear, otherwise it is nonlinear. 

(c) The system has a solution on $I$ :$\alpha <t<\beta$ if there exists $n$ functions $x_1=f_1(t)$, $x_2=f_2(t)$,…,$x_n=f_n(t)$ that are differentiable on $I$ and satisfies the system of equations at all points $t$ in $I$. 

(d) Initial conditions may also be prescribed to give an IVP: $x_1(t_0)=b_1$, $x_2(t_0)=b_2$,…,$x_n(t_0)=b_n$. 

Theorem: Suppose $F_1,...,F_n$ and $\partial F_1/\partial x_1$, $\partial F_1/\partial x_2$,…,$\partial F_1/\partial x_n$, $\partial F_2/\partial x_1$,…,$\partial F_n/\partial x_n$ are continuous in the region $R$ of $t,x_1,...,x_n$-space defined by $\alpha <t<\beta$, ${\alpha }_1<x_1<{\beta }_1$, …,${\alpha }_n<x_n<{\beta }_n$ and let the point $(t_0,b_1,...,b_n)$ be contained in $R$. Then in some interval $(t_0-h,t_0+h)$ there exists a unique solution $x_1=f_1(t),x_2=f_2(t),...,x_n=f_n(t)$ that satisfies the IVP: $x_1(t_0)=b_1$, $x_2(t_0)=b_2$,…,$x_n(t_0)=b_n$,

$$\begin{array}{rl}{x}_1^{\prime }& =F_1(t,x_1,x_2,\dots ,x_n)\\ x_{2\prime }& =F_2(t,x_1,x_2,\dots ,x_n)\\ ⋮& =⋮\\ x_n^{\prime }& =F_2(t,x_1,x_2,\dots ,x_n).\end{array}$$

The focus of this chapter is the linear system and $n=2$.  When $F_i^{\prime }s$ are a linear functions, we can rewrite the system of equations:

$$\begin{array}{rl}{x}_1^{\prime }& =p_{11}(t)x_1+p_{12}(t)x_2+\dots .+p_{1n}(t)x_n+g_1(t)\\ x_{2\prime }& =p_{21}(t)x_1+p_{22}(t)x_2+\dots .+p_{2n}(t)x_n+g_2(t)\\ ⋮& =⋮\\ x_n^{\prime }& =p_{n1}(t)x_1+p_{n2}(t)x_2+\dots .+p_{nn}(t)x_n+g_n(t).\end{array}$$

If each of the $g_i(t)$ is zero on the interval $I$, then the system is homogeneous, otherwise it is nonhomogeneous.

Example 1: Transform the ODE into a system of first order equations. $y^{″}+2y^{\prime }+3y=0$.

Exercise 1: Transform the ODE into a system of first order equations. $y^{″}+3y^{\prime }-4y=0$.

Example 2: Transform the IVP into a system of first order equations. $y^{″}+2y^{\prime }-3y=\text{sin}(4t)$, $y(0)=3$, $y^{\prime }(0)=2$.

Exercise 2: Transform the IVP into a system of first order equations. $y^{″}-2y^{\prime }-8y=\text{cos}(5t)$, $y(0)=-2$, $y^{\prime }(0)=4$.

Example 3: Transform the IVP into a system of first order equations. $y^{″}+2\text{cos}(t)y^{\prime }-e^{2t}y=\text{sin}(t)$, $y(0)=b_1$, $y^{\prime }(0)=b_2$.

Exercise 3: Transform the IVP into a system of first order equations. $y^{″}+e^{3t}{y}^{\prime }-\text{sin}(2t)y=t^2+2$, $y(0)=c$, $y^{\prime }(0)=d$.

Example 4: Transform the system of first order equations into one ODE.  $x_1^{\prime }=-2x_1+x_2$, $x_2^{\prime }=x_1-2x_2$, $x_1(0)=2$, $x_2(0)=4$. 

Exercise 4: Transform the system of first order equations into one 2nd oder ODE. $x_1^{\prime }=3x_1-2x_2$, $x_2^{\prime }=2x_1-x_2$, $x_1(0)=3$, $x_2(0)=1$. 

Group Work

1. Transform the system of first order equations into one 2nd order IVP. Then solve the IVP, and use it to find $x_1$ and $x_2$. $x_1^{\prime }=5x_1-x_2$, $x_2^{\prime }=3x_1+x_2$, $x_1(0)=0$, $x_2(0)=3$. 

2. Transform the system of first order equations into one 2nd order IVP. Then solve the IVP, and use it to find $x_1$ and $x_2$. $x_1^{\prime }=-2x_1$, $x_2^{\prime }=5x_1-3x_2$, $x_1(0)=0$, $x_2(0)=1$. 

3. Transform the IVP into a system of first order equations. $y^{″}+(t^2+2)y^{\prime }-y=e^t$, $y(0)=1$, $y^{\prime }(0)=-1$.