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

$$\begin{align*} x_{1}’ & =F_{1}(t,x_{1},x_{2},…,x_{n})\\ x_{2′} & =F_{2}(t,x_{1},x_{2},…,x_{n})\\ \vdots & =\vdots\\ x_{n}’ & =F_{2}(t,x_{1},x_{2},…,x_{n}) \end{align*}$$

 where each [latex]x_{i}[/latex] is a function of [latex]t[/latex]. 

(b) If each [latex]F_{i}[/latex] is a linear function of [latex]x_{1},...,x_{n}[/latex], then the system of equations is said to be linear, otherwise it is nonlinear. 

(c) The system has a solution on [latex]I[/latex] :[latex]\alpha \lt t \lt \beta[/latex] if there exists [latex]n[/latex] functions [latex]x_{1}=f_{1}(t)[/latex], [latex]x_{2}=f_{2}(t)[/latex],…,[latex]x_{n}=f_{n}(t)[/latex] that are differentiable on [latex]I[/latex] and satisfies the system of equations at all points [latex]t[/latex] in [latex]I[/latex]. 

(d) Initial conditions may also be prescribed to give an IVP: [latex]x_{1}(t_{0})=b_{1}[/latex], [latex]x_{2}(t_{0})=b_{2}[/latex],…,[latex]x_{n}(t_{0})=b_{n}[/latex]. 

Theorem: Suppose [latex]F_{1},...,F_{n}[/latex] and [latex]\partial F_{1}/\partial x_{1}[/latex], [latex]\partial F_{1}/\partial x_{2}[/latex],…,[latex]\partial F_{1}/\partial x_{n}[/latex], [latex]\partial F_{2}/\partial x_{1}[/latex],…,[latex]\partial F_{n}/\partial x_{n}[/latex] are continuous in the region [latex]R[/latex] of [latex]t,x_{1},...,x_{n}[/latex]-space defined by [latex]\alpha \lt t \lt \beta[/latex], [latex]\alpha_{1} \lt x_{1} \lt \beta_{1}[/latex], …,[latex]\alpha_{n} \lt x_{n} \lt \beta_{n}[/latex] and let the point [latex](t_{0},b_{1},...,b_{n})[/latex] be contained in [latex]R[/latex]. Then in some interval [latex](t_{0}-h,t_{0}+h)[/latex] there exists a unique solution [latex]x_{1}=f_{1}(t),x_{2}=f_{2}(t),...,x_{n}=f_{n}(t)[/latex] that satisfies the IVP: [latex]x_{1}(t_{0})=b_{1}[/latex], [latex]x_{2}(t_{0})=b_{2}[/latex],…,[latex]x_{n}(t_{0})=b_{n}[/latex],

$$\begin{align*} x_{1}’ & =F_{1}(t,x_{1},x_{2},…,x_{n})\\ x_{2′} & =F_{2}(t,x_{1},x_{2},…,x_{n})\\ \vdots & =\vdots\\ x_{n}’ & =F_{2}(t,x_{1},x_{2},…,x_{n}). \end{align*}$$

The focus of this chapter is the linear system and [latex]n=2[/latex].  When [latex]F_{i}'s[/latex] are a linear functions, we can rewrite the system of equations:

$$\begin{align*} x_{1}’ & =p_{11}(t)x_{1}+p_{12}(t)x_{2}+….+p_{1n}(t)x_{n}+g_{1}(t)\\ x_{2′} & =p_{21}(t)x_{1}+p_{22}(t)x_{2}+….+p_{2n}(t)x_{n}+g_{2}(t)\\ \vdots & =\vdots\\ x_{n}’ & =p_{n1}(t)x_{1}+p_{n2}(t)x_{2}+….+p_{nn}(t)x_{n}+g_{n}(t). \end{align*}$$

If each of the [latex]g_{i}(t)[/latex] is zero on the interval [latex]I[/latex], then the system is homogeneous, otherwise it is nonhomogeneous.

Example 1: Transform the ODE into a system of first order equations. [latex]y''+2y'+3y=0[/latex].

Exercise 1: Transform the ODE into a system of first order equations. [latex]y''+3y'-4y=0[/latex].

Example 2: Transform the IVP into a system of first order equations. [latex]y''+2y'-3y=\text{sin}(4t)[/latex], [latex]y(0)=3[/latex], [latex]y'(0)=2[/latex].

Exercise 2: Transform the IVP into a system of first order equations. [latex]y''-2y'-8y=\text{cos}(5t)[/latex], [latex]y(0)=-2[/latex], [latex]y'(0)=4[/latex].

Example 3: Transform the IVP into a system of first order equations. [latex]y''+2\text{cos}(t)y'-e^{2t}y=\text{sin}(t)[/latex], [latex]y(0)=b_{1}[/latex], [latex]y'(0)=b_{2}[/latex].

Exercise 3: Transform the IVP into a system of first order equations. [latex]y''+e^{3t}y'-\text{sin}(2t)y=t^{2}+2[/latex], [latex]y(0)=c[/latex], [latex]y'(0)=d[/latex].

Example 4: Transform the system of first order equations into one ODE.  [latex]x_{1}'=-2x_{1}+x_{2}[/latex], [latex]x_{2}'=x_{1}-2x_{2}[/latex], [latex]x_{1}(0)=2[/latex], [latex]x_{2}(0)=4[/latex]. 

Exercise 4: Transform the system of first order equations into one 2nd oder ODE. [latex]x_{1}'=3x_{1}-2x_{2}[/latex], [latex]x_{2}'=2x_{1}-x_{2}[/latex], [latex]x_{1}(0)=3[/latex], [latex]x_{2}(0)=1[/latex]. 

Group Work

1. Transform the system of first order equations into one 2nd order IVP. Then solve the IVP, and use it to find [latex]x_{1}[/latex] and [latex]x_{2}[/latex]. [latex]x_{1}'=5x_{1}-x_{2}[/latex], [latex]x_{2}'=3x_{1}+x_{2}[/latex], [latex]x_{1}(0)=0[/latex], [latex]x_{2}(0)=3[/latex]. 

2. Transform the system of first order equations into one 2nd order IVP. Then solve the IVP, and use it to find [latex]x_{1}[/latex] and [latex]x_{2}[/latex]. [latex]x_{1}'=-2x_{1}[/latex], [latex]x_{2}'=5x_{1}-3x_{2}[/latex], [latex]x_{1}(0)=0[/latex], [latex]x_{2}(0)=1[/latex]. 

3. Transform the IVP into a system of first order equations. [latex]y''+(t^{2}+2)y'-y=e^{t}[/latex], [latex]y(0)=1[/latex], [latex]y'(0)=-1[/latex].