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10.2E: Linear Systems of Differential Equations (Exercises)

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Q10.2.1

1. Rewrite the system in matrix form and verify that the given vector function satisfies the system for any choice of the constants c1 and c2.

  1. y1=2y1+4y2y2=4y1+2y2;y=c1[11]e6t+c2[11]e2t
  2. y1=2y12y2y2=5y1+2y2;y=c1[11]e4t+c2[25]e3t
  3. y1=4y110y2y2=3y1+17y2;y=c1[53]e2t+c2[21]et
  4. y1=2y1+2y2y2=2y1+2y2;y=c1[11]e3t+c2[11]et

2. Rewrite the system in matrix form and verify that the given vector function satisfies the system for any choice of the constants c1, c2, and c3.

  1. y1=y1+2y2+3y3y2=y2+6y3y3=2y3;
    y=c1[110]et+c2[100]et+c3[121]e2t
  2. y1=2y1+2y2+2y3y2=2y1+2y2+2y3y3=2y1+2y2;+2y3
    y=c1[101]e2t+c2[011]e2t+c3[111]e4t
  3. y1=y1+2y2+2y3y2=2y12y2+2y3y3=2y1+2y22y3;
    y=c1[101]e3t+c2[011]e3t+c3[111]e3t
  4. y1=3y12y22y3y2=2y1+3y2+2y3y3=4y13y22y3;
    y=c1[101]e2t+c2[111]e3t+c3[137]et

3. Rewrite the initial value problem in matrix form and verify that the given vector function is a solution.

  1. y1=2y1+4y2y2=2y1+4y2,y1(0)=1y2(0)=0; y=2[11]e2t[12]e3t
  2. y1=5y1+3y2y2=y1+y2,y1(0)=12y2(0)=6; y=3[11]e2t+3[31]e4t

4. Rewrite the initial value problem in matrix form and verify that the given vector function is a solution.

  1. y1=6y1+4y2+4y3y2=7y12y2y3,y3=7y1+4y2+3y3,y1(0)=3y2(0)=6y3(0)=4
    y=[111]e6t+2[121]e2t+[011]et
  2. y1=8y1+7y2+17y3y2=5y16y219y3,y3=5y1+7y2+10y3, y1(0)=2y2(0)=4y3(0)=3
    y=[111]e8t+[011]e3t+[121]et

5. Rewrite the system in matrix form and verify that the given vector function satisfies the system for any choice of the constants c1 and c2.

  1. y1=3y1+2y2+32ty2=5y1+3y2+63t
    y=c1[2cost3costsint]+c2[2sint3sint+cost]+[1t]
  2. y1=3y1+y25ety2=y1+y2+et
    y=c1[11]e2t+c2[1+tt]e2t+[13]et
  3. y1=y14y2+4et+8tety2=y14y2+e3t+(4t+2)et
    y=c1[21]e3t+c2[21]et+[e3t2tet]
  4. y1=6y13y2+14e2t+12ety2=6y12y2+7e2t12et
    y=c1[31]e5t+c2[11]e3t+[e2t+3et2e2t3et]

6. Convert the linear scalar equation

P0(t)y(n)+P1(t)y(n1)++Pn(t)y(t)=F(t)

into an equivalent n×n system

y=A(t)y+f(t),

and show that A and f are continuous on an interval (a,b) if and only if (A) is normal on (a,b).

7. A matrix function

Q(t)=[q11(t)q12(t)q1s(t)q21(t)q22(t)q2s(t)qr1(t)qr2(t)qrs(t)]

is said to be differentiable if its entries {qij} are differentiable. Then the derivative Q is defined by

Q(t)=[q11(t)q12(t)q1s(t)q21(t)q22(t)q2s(t)qr1(t)qr2(t)qrs(t)]

  1. Prove: If P and Q are differentiable matrices such that P+Q is defined and if c1 and c2 are constants, then (c1P+c2Q)=c1P+c2Q.
  2. Prove: If P and Q are differentiable matrices such that PQ is defined, then (PQ)=PQ+PQ.

8. Verify that Y=AY.

  1. Y=[e6te2te6te2t],A=[2442]
  2. Y=[e4t2e3te4t5e3t],A=[2251]
  3. Y=[5e2t2et3e2tet],A=[41037]
  4. Y=[e3tete3tet],A=[2112]
  5. Y=[etete2tet02e2t00e2t],A=[123016002]
  6. Y=[e2te2te4t0e2te4te2t0e4t],A=[022202220]
  7. Y=[e3te3t0e3t0e3te3te3te3t],A=[966636663]
  8. Y=[e2te3tet0e3t3ete2te3t7et],A=[311232412]

9. Suppose

y1=[y11y21]andy2=[y12y22]

are solutions of the homogeneous system

y=A(t)y,

and define

Y=[y11y12y21y22].

  1. Show that Y=AY.
  2. Show that if c is a constant vector then y=Yc is a solution of (A).
  3. State generalizations of (a) and (b) for n×n systems.

10. Suppose Y is a differentiable square matrix.

  1. Find a formula for the derivative of Y2.
  2. Find a formula for the derivative of Yn, where n is any positive integer.
  3. State how the results obtained in (a) and (b) are analogous to results from calculus concerning scalar functions.

11. It can be shown that if Y is a differentiable and invertible square matrix function, then Y1 is differentiable.

  1. Show that (Y1)=Y1YY1. (Hint: Differentiate the identity Y1Y=I.)
  2. Find the derivative of Yn=(Y1)n, where n is a positive integer.
  3. State how the results obtained in (a) and (b) are analogous to results from calculus concerning scalar functions.

12. Show that Theorem 10.2.1 implies Theorem 9.1.1. HINT: Write the scalar function P0(x)y(n)+P1(x)y(n1)++Pn(x)y=F(x) as an n×n system of linear equations.

13. Suppose y is a solution of the n×n system y=A(t)y on (a,b), and that the n×n matrix P is invertible and differentiable on (a,b). Find a matrix B such that the function x=Py is a solution of x=Bx on (a,b).


This page titled 10.2E: Linear Systems of Differential Equations (Exercises) is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by William F. Trench.

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