Download to read offline
Uploaded byRendy Robert
Ch07 1
This document introduces systems of first order linear differential equations. It defines such systems as having the general form where each derivative is a linear combination of the variables. It describes how higher order equations can be converted into equivalent first order systems by defining new variables for each derivative. The document states that for an initial value problem with a linear system, there exists a unique solution defined in some interval if the coefficients are continuous.
Ch07 1
- 1. Ch 7.1: Introductionto Systems of First Order Linear Equations A system of simultaneous first order ordinary differential equations has the general form where each xk is a function of t. If each Fk is a linear function of x1, x2, …, xn, then the system of equations is said to be linear, otherwise it is nonlinear. Systems of higher order differential equations can similarly be defined. ),,,( ),,,( ),,,( 21 2122 2111 nnn n n xxxtFx xxxtFx xxxtFx =′ =′ =′
- 2. Example 1 The motionof a spring-mass system from Section 3.8 was described by the equation This second order equation can be converted into a system of first order equations by letting x1= u and x2 = u'. Thus or 0)(192)(16)( =+′+′′ tututu 019216 122 21 =++′ =′ xxx xx 122 21 19216 xxx xx −−=′ =′
- 3. Nth Order ODEsand Linear 1st Order Systems The method illustrated in previous example can be used to transform an arbitrary nth order equation into a system of n first order equations, first by defining Then ( ))1()( ,,,,, − ′′′= nn yyyytFy )1( 321 ,,,, − =′′=′== n n yxyxyxyx ),,,( 21 1 32 21 nn nn xxxtFx xx xx xx =′ =′ =′ =′ −
- 4. Solutions of FirstOrder Systems A system of simultaneous first order ordinary differential equations has the general form It has a solution on I: α < t < β if there exists n functions that are differentiable on I and satisfy the system of equations at all points t in I. Initial conditions may also be prescribed to give an IVP: ).,,,( ),,,( 21 2111 nnn n xxxtFx xxxtFx =′ =′ )(,),(),( 2211 txtxtx nn φφφ === 0 0 0 202 0 101 )(,,)(,)( nn xtxxtxxtx ===
- 5. Example 2 The equation canbe written as system of first order equations by letting x1= y and x2 = y'. Thus A solution to this system is which is a parametric description for the unit circle. π20,0 <<=+′′ tyy 12 21 xx xx −=′ =′ π20),cos(),sin( 21 <<== ttxtx
- 6. Theorem 7.1.1 Suppose F1,…,Fn and ∂F1/∂x1,…, ∂F1/∂xn,…, ∂Fn/∂ x1,…, ∂Fn/∂xn, are continuous in the region R of t x1x2…xn-space defined by α < t < β, α1 < x1 < β1, …, αn < xn < βn, and let the point be contained in R. Then in some interval (t0 - h, t0+ h) there exists a unique solution that satisfies the IVP. ( )00 2 0 10 ,,,, nxxxt )(,),(),( 2211 txtxtx nn φφφ === ),,,( ),,,( ),,,( 21 2122 2111 nnn n n xxxtFx xxxtFx xxxtFx =′ =′ =′
- 7. Linear Systems If eachFk is a linear function of x1, x2, …, xn, then the system of equations has the general form If each of the gk(t) is zero on I, then the system is homogeneous, otherwise it is nonhomogeneous. )()()()( )()()()( )()()()( 2211 222221212 112121111 tgxtpxtpxtpx tgxtpxtpxtpx tgxtpxtpxtpx nnnnnnn nn nn ++++=′ ++++=′ ++++=′
- 8. Theorem 7.1.2 Suppose p11,p12,…, pnn, g1,…, gn are continuous on an interval I: α < t < β with t0in I, and let prescribe the initial conditions. Then there exists a unique solution that satisfies the IVP, and exists throughout I. 00 2 0 1 ,,, nxxx )(,),(),( 2211 txtxtx nn φφφ === )()()()( )()()()( )()()()( 2211 222221212 112121111 tgxtpxtpxtpx tgxtpxtpxtpx tgxtpxtpxtpx nnnnnnn nn nn ++++=′ ++++=′ ++++=′