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.
- y′1=2y1+4y2y′2=4y1+2y2;y=c1[11]e6t+c2[1−1]e−2t
- y′1=−2y1−2y2y′2=−5y1+2y2;y=c1[11]e−4t+c2[−25]e3t
- y′1=−4y1−10y2y′2=3y1+17y2;y=c1[−53]e2t+c2[2−1]et
- y′1=2y1+2y2y′2=2y1+2y2;y=c1[11]e3t+c2[1−1]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.
- y′1=−y1+2y2+3y3y′2=y2+6y3y′3=−2y3;
y=c1[110]et+c2[100]e−t+c3[1−21]e−2t - y′1=2y1+2y2+2y3y′2=2y1+2y2+2y3y′3=2y1+2y2;+2y3
y=c1[−101]e−2t+c2[0−11]e−2t+c3[111]e4t - y′1=−y1+2y2+2y3y′2=2y1−2y2+2y3y′3=2y1+2y2−2y3;
y=c1[−101]e−3t+c2[0−11]e−3t+c3[111]e3t - y′1=3y1−2y2−2y3y′2=−2y1+3y2+2y3y′3=−4y1−3y2−2y3;
y=c1[101]e2t+c2[1−11]e3t+c3[1−37]e−t
3. Rewrite the initial value problem in matrix form and verify that the given vector function is a solution.
- y′1=−2y1+4y2y′2=−2y1+4y2,y1(0)=1y2(0)=0; y=2[11]e2t−[12]e3t
- y′1=5y1+3y2y′2=−y1+y2,y1(0)=12y2(0)=−6; y=3[1−1]e2t+3[3−1]e4t
4. Rewrite the initial value problem in matrix form and verify that the given vector function is a solution.
- y′1=6y1+4y2+4y3y′2=−7y1−2y2−y3,y′3=7y1+4y2+3y3,y1(0)=3y2(0)=−6y3(0)=4
y=[1−11]e6t+2[1−21]e2t+[0−11]e−t - y′1=−8y1+7y2+17y3y′2=−5y1−6y2−19y3,y′3=−5y1+7y2+10y3, y1(0)=2y2(0)=−4y3(0)=3
y=[1−11]e8t+[0−11]e3t+[1−21]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.
- y′1=−3y1+2y2+3−2ty′2=−5y1+3y2+6−3t
y=c1[2cost3cost−sint]+c2[2sint3sint+cost]+[1t] - y′1=3y1+y2−5ety′2=−y1+y2+et
y=c1[−11]e2t+c2[1+t−t]e2t+[13]et - y′1=−y1−4y2+4et+8tety′2=−y1−4y2+e3t+(4t+2)et
y=c1[21]e−3t+c2[−21]et+[e3t2tet] - y′1=−6y1−3y2+14e2t+12ety′2=6y1−2y2+7e2t−12et
y=c1[−31]e−5t+c2[−11]e−3t+[e2t+3et2e2t−3et]
6. Convert the linear scalar equation
P0(t)y(n)+P1(t)y(n−1)+⋯+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)=[q′11(t)q′12(t)⋯q′1s(t)q′21(t)q′22(t)⋯q′2s(t)⋮⋮⋱⋮q′r1(t)q′r2(t)⋯q′rs(t)]
- 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′.
- Prove: If P and Q are differentiable matrices such that PQ is defined, then (PQ)′=P′Q+PQ′.
8. Verify that Y′=AY.
- Y=[e6te−2te6t−e−2t],A=[2442]
- Y=[e−4t−2e3te−4t5e3t],A=[−2−2−51]
- Y=[−5e2t2et3e2t−et],A=[−4−1037]
- Y=[e3tete3t−et],A=[2112]
- Y=[ete−te−2tet0−2e−2t00e−2t],A=[−12301600−2]
- Y=[−e−2t−e−2te4t0e−2te4te−2t0e4t],A=[022202220]
- Y=[e3te−3t0e3t0−e−3te3te−3te−3t],A=[−966−636−663]
- Y=[e2te3te−t0−e3t−3e−te2te3t7e−t],A=[3−1−1−2324−1−2]
9. Suppose
y1=[y11y21]andy2=[y12y22]
are solutions of the homogeneous system
y′=A(t)y,
and define
Y=[y11y12y21y22].
- Show that Y′=AY.
- Show that if c is a constant vector then y=Yc is a solution of (A).
- State generalizations of (a) and (b) for n×n systems.
10. Suppose Y is a differentiable square matrix.
- Find a formula for the derivative of Y2.
- Find a formula for the derivative of Yn, where n is any positive integer.
- 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 Y−1 is differentiable.
- Show that (Y−1)′=−Y−1Y′Y−1. (Hint: Differentiate the identity Y−1Y=I.)
- Find the derivative of Y−n=(Y−1)n, where n is a positive integer.
- 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(n−1)+⋯+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).