
# 10.2E: Linear Systems of Differential Equations (Exercises)

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into an equivalent $$n\times n$$ system

${\bf y'}=A(t){\bf y}+{\bf f}(t),$

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

[exer:10.2.7] A matrix function

$Q(t)=\matfunc qrs$

is said to be differentiable if its entries $$\{q_{ij}\}$$ are differentiable. Then the derivative $$Q'$$ is defined by

$Q'(t)=\matfunc {q'}rs.$

Prove: If $$P$$ and $$Q$$ are differentiable matrices such that $$P+Q$$ is defined and if $$c_1$$ and $$c_2$$ are constants, then

$(c_1P+c_2Q)'=c_1P'+c_2Q'.$

Prove: If $$P$$ and $$Q$$ are differentiable matrices such that $$PQ$$ is defined, then

$(PQ)'=P'Q+PQ'.$

[exer:10.2.8] Verify that $$Y' = AY$$.

$$\{Y = \twobytwo {e^{6t}}{e^{-2t}} {e^{6t}}{-e^{-2t}}, \quad A = \twobytwo 2 4 4 2}$$

$$\{Y = \twobytwo {e^{-4t}} {-2e^{3t}} {e^{-4t}} {5e^{3t}}, \quad A = \twobytwo {-2} {-2} {-5} {1}}$$

$$\{Y = \twobytwo {-5e^{2t}} {2e^t} {3e^{2t}} {-e^t}, \quad A = \twobytwo {-4} {-10} 3 7}$$

$$\{Y = \twobytwo {e^{3t}} {e^t} {e^{3t}} {-e^t}, \quad A = \twobytwo 2 1 1 2}$$

$$Y = \left[\begin{array}{crr} e^t&e^{-t}& e^{-2t}\\ e^t&0&-2e^{-2t}\\ 0&0&e^{-2t}\end{array}\right], \quad A = \threebythree {-1} 2 {3} {0} 1 6 0 0 {-2}$$

$$\{Y = \cthreebythree {-e^{-2t}} {-e^{-2t}} {e^{4t}} 0 {\phantom{-} e^{-2t}} {e^{4t}} {e^{-2t}} 0 {e^{4t}}, \quad A = \threebythree 0 2 2 2 0 2 2 2 0}$$

$$\{Y = \cthreebythree {e^{3t}} {e^{-3t}} 0 {e^{3t}} 0 {-e^{-3t}} {e^{3t}} {e^{-3t}} {\phantom{-}e^{-3t}}, \quad A = \threebythree {-9}66{-6}36{-6}63}$$

$$Y = \left[\begin{array}{crr} e^{2t}&e^{3t}& e^{-t}\\ 0&-e^{3t}&-3e^{-t}\\ e^{2t}&e^{3t}&7e^{-t}\end{array}\right] , \quad A = \threebythree 3 {-1} {-1}{-2} 3 2 4 {-1} {-2}$$

[exer:10.2.9] Suppose

${\bf y}_1=\twocol{y_{11}}{y_{21}}\mbox{\quad and \quad}{\bf y}_2=\twocol{y_{12}}{y_{22}}$

are solutions of the homogeneous system

${\bf y}'=A(t){\bf y}, \eqno{\rm (A)}$

and define

$Y= \twobytwo{y_{11}}{y_{12}}{y_{21}}{y_{22}}.$

Show that $$Y'=AY$$.

Show that if $${\bf c}$$ is a constant vector then $${\bf y}= Y{\bf c}$$ is a solution of (A).

State generalizations of

and

## b

for $$n\times n$$ systems.

[exer:10.2.10] Suppose $$Y$$ is a differentiable square matrix.

Find a formula for the derivative of $$Y^2$$.

Find a formula for the derivative of $$Y^n$$, where $$n$$ is any positive integer.

State how the results obtained in

and

## b

are analogous to results from calculus concerning scalar functions.

[exer:10.2.11] It can be shown that if $$Y$$ is a differentiable and invertible square matrix function, then $$Y^{-1}$$ is differentiable.

Show that ($$Y^{-1})'= -Y^{-1}Y'Y^{-1}$$. (Hint: Differentiate the identity $$Y^{-1}Y=I$$.)

Find the derivative of $$Y^{-n}=\left(Y^{-1}\right)^n$$, where $$n$$ is a positive integer.

State how the results obtained in

and

## b

are analogous to results from calculus concerning scalar functions.

[exer:10.2.12] Show that Theorem [thmtype:10.2.1} implies Theorem [thmtype:9.1.1}.

[exer:10.2.13] Suppose $${\bf y}$$ is a solution of the $$n\times n$$ system $${\bf y}'=A(t){\bf y}$$ on $$(a,b)$$, and that the $$n\times n$$ matrix $$P$$ is invertible and differentiable on $$(a,b)$$. Find a matrix $$B$$ such that the function $${\bf x}=P{\bf y}$$ is a solution of $${\bf x}'=B{\bf x}$$ on $$(a,b)$$.