MATH 312 Section 8.3: Non-homogeneous Systems

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1 MATH 32 Section 8.3: Non-homogeneous Systems Prof. Jonathan Duncan Walla Walla College Spring Quarter, 2007

2 Outline Undetermined Coefficients 2 Variation of Parameter 3 Conclusions

3 Undetermined Coefficients In section 4.4, which we skipped, the method of undetermined coefficients was introduced without annihilators. Form of a Particular Solution To determine the form of a particular solution to the non-homogeneous differential equation d n y dx n + a n (x) d n y dx n + + a n (x) dy dx + a n(x)y = g(x) we examine the form of g(x). Determine the form of a particular solution y p (x) to the differential equation above if g(x) is 5x cos 4x 3 x 2 e 5x

4 Applications to Matrices If we apply this procedure to a system of differential equations in matrix form, we see much the same phenomena. Form of a Particular Solution To determine the form of a particular solution to the non-homogeneous system of equations X = A X + F (t) we examine the form of F (t). Determine the general form of a particular solution X p if F (t) is 2 7 [ 2 ] e t + 0 t + 5 7

5 Solution Method With this guessing technique we can now wolve non-homogeneous systems using the following method. Solution Method To solve a non-homogeneous system by undetermined coefficients Solve the associated homogeneous system to find X c. Determine the form of X p with undetermined coefficients. Find X p by plugging the form above into X = A X + F. The solution is then X = X c + X p Solve the initial value problem below subject to X (0) = X = X

6 Another We finish this section looking at another example. Find the general solution to the differential equation dx = 5x +9y + 2t+ dt dy dt = x+y + 6 Solve the system of differential equations X = X + e 4t

7 Variation of Parameter While the method of undetermined coefficients is straight-forward, it does have some limits. Limitations of Undetermined Coefficients Undetermined coefficients is not an appropriate method when More complicated F (t) s are much harder to deal with. When the complementary solution has eigenvalues with multiplicity, this method may not work. In section 4.6 we developed the method of variation of parameter for solving a single DE based on the assumption that for some u (x) and u 2 (x). y p = u (x)y (x) + u 2 (x)y 2 (x)

8 The Fundamental Matrix To see how this method would generalize to a system of differential equations, consider the following. Fundamental Matrix If X, X 2,..., X n form a fundamental set of solutions to the homogeneous system X = AX on some interval I, then the general solution can be expressed as x x n x x 2 x n c x 2 X = c c x 2n n.. = x 2 x 22 x 2n c = ϕ(t) C x n x nn x n x n2 x nn c n

9 Deriving the Formula We assume then that the form of the particular solution is X p = ϕ(t) U(t) for some U(t). Properties of ϕ(t) Note that ϕ(t) has two important properties. ϕ(t) is non-singular and thus invertible. ϕ (t) = Aϕ(t). Formula for X p Using the properties and assumptions above, show that X p = ϕ(t) ϕ (t) F (t) dt

10 s We now apply the variation of parameter formula to solve the following examples. Solve the system of equations dx = 2y +2 dt dy = 3y x + e 3t dt X = ϕ(t) C + ϕ(t) ϕ (t) F (t) dt X = c e 2t 2 + c 2 [ ] e t

11 Initial Value s The variation of parameter formula can be used to solve initial value problems with a slight adjustment. Initial Value Formula Using the fundamental theorem of calculus, show that the solution to the initial value problem X = A X + F (t) subject to X (t 0 ) = ϕ(t 0 ) C = X 0 is t X = ϕ(t)ϕ (t 0 ) X 0 + ϕ(t) ϕ (s) F (s) ds t 0 Solve the initial value problem below subject to X (0) = X = 3 X + 3 4e 2t 4e 4t.

12 Important Concepts Things to Remember from Section 8.3 Using undetermined coefficients with matrix equations. 2 Solving non-homogeneous systems using undetermined coefficients. 3 Finding the fundamental matrix for a non-homogeneous system. 4 Using variation of parameters to solve a non-homogeneous system.