Chapter 6.7

Front 1

General solution for nonhomogeneous linear systems by undetermined coefficients with constant forcing

Back 1

for x'=Ax+f(t) the particular solution is x'=Ax+b

where b is a constant vector, we try a constant xp, which gives xp'=0. Substituting xp and xp'=0 into the equation gives, Axp +b=0 or xp =-A⁻¹b

so that the total general solution would be

x=xh - A⁻¹b

Front 2

When, generally, is the method of undetermined coefficients applied

Back 2

Whenever A is a matrix of constant coefficients and the forcing vector f(t) is restricted to the function families in Sec. 4.4

Front 3

Function families of undetermined coefficients

Back 3

1. polynomials in t

2. e^(αt)

3. cos(kt), sin(kt)

4. finite sums and products of the above functions

Front 4

When to use variation of parameters for nonhomogeneous linear systems

Back 4

When the elements of the matrix A(t) can be functions of t

Front 5

Steps of variation of parameters for nonhomogeneous linear systems

Back 5

The funadmental matrix for the homogeneous system is a matrix X(t) where X'(t)=A(t)X(t),

2. X(T) is given by xh =X(t)c where c is an arbitrary constant vector

3. If we replace c with a vector function v(t) we get xp= X(t)v(t)

4. Plug it into the original equation (Xv)' = AXv + f

5. solve for v' so that v'= X⁻¹f

6. v= ∫X⁻¹(t)f(t)dt+k

7. set k= {0,0} so that xp= X(t)∫X⁻¹(t)f(t)dt

8. Then we get the general solution

x(t)= X(t)c +X(t)∫X⁻¹(t)f(t)dt

Front 6

What does c equal if x(0)=x₀ for variation of parameters general solution

Back 6

c= X⁻¹(0)x₀

Front 7

General solution of x'=A(t)x+ft

Back 7

x(t)= X(t)X⁻¹(0)x₀ + X(t) ∫X⁻¹(s)f(s)ds