In , assume that no solution flows out of the system from

Problem 38E Arms Race. A simplified mathematical model for an arms race between two countries whose expenditures for defense are expressed by the variables x(t) and y(t) is given by the linear system where a and b are constants that measure the trust (or distrust) each country has for the other. Determine whether there is going to be disarmament (x and y approach 0 as t increases), a stabilized arms race (x and y approach a constant as or a runaway arms race (x and y approach + ? as

Problem 1E Let A=D-1,B=D+2,C=D2+D-2, Where D=d/dt. For y = t3-8, compute (a) A[y] (b) B[A[y]] (c) B[y] (d) A[B[y]] (e) C[y]

In Problems 3-18, use the elimination method to find a general solution for the given linear system, where differentiation is with respect to . \(x^{\prime}+y^{\prime}-x=5\), \(x^{\prime}+y^{\prime}+y=1\) Equation transcription: Text transcription: x^{prime}+y^{prime}-x=5 x^{prime}+y^{prime}+y=1

In Problems 3-18, use the elimination method to find a general solution for the given linear system, where differentiation is with respect to . \(x^{\prime}=x-y\), \(x^{\prime}-y^{\prime}=0 y^{\prime}=y-4 x\) Equation transcription: Text transcription: x^{prime}=x-y x^{prime}-y^{prime}=0 y^{prime}=y-4 x

In Problems 3-18, use the elimination method to find a general solution for the given linear system, where differentiation is with respect to . \(x^{\prime}=3 x-2 y+\sin t\) \(y^{\prime}=4 x-y-\cos t\) Equation Transcription: Text Transcription: x'=3x-2y+sin? t y'=4x-y-cos? t

Problem 3E In Problems 3–18, use the elimination method to find a general solution for the given linear system, where differentiation is with respect to t.

Problem 7E In Problems 3–18, use the elimination method to find a general solution for the given linear system, where differentiation is with respect to t.

In Problems 3-18, use the elimination method to find a general solution for the given linear system, where differentiation is with respect to . \(x^{\prime}+y^{\prime}+2 x=0\) \(x^{\prime}+y^{\prime}-x-y=\sin t\) Equation Transcription: Text Transcription: x'+y'+2x=0 x'+y'-x-y=sin? t

In Problems 3-18, use the elimination method to find a general solution for the given linear system, where differentiation is with respect to . \((D-3)[x]+(D-1)[y]=t\), \((D+1)[x]+(D+4)[y]=1\) Equation transcription: Text transcription: (D-3)[x]+(D-1)[y]=t (D+1)[x]+(D+4)[y]=1

In Problems 3-18, use the elimination method to find a general solution for the given linear system, where differentiation is with respect to . \(2 x^{\prime}+y^{\prime}-x-y=e^{-t}\), \(x^{\prime}+y^{\prime}+2 x+y=e^{t}\) Equation transcription: Text transcription: 2 x^{prime}+y^{prime}-x-y=e^{-t} x^{prime}+y^{prime}+2 x+y=e^{t}

Problem 11E In Problems 3–18, use the elimination method to find a general solution for the given linear system, where differentiation is with respect to t.

Problem 12E In Problems 3–18, use the elimination method to find a general solution for the given linear system, where differentiation is with respect to t.

Problem 13E In Problems 3–18, use the elimination method to find a general solution for the given linear system, where differentiation is with respect to t.

In Problems 3-18, use the elimination method to find a general solution for the given linear system, where differentiation is with respect to . \(\frac{d x}{d t}+y=t^{2}\), \(-x+\frac{d y}{d t}=1\) Equation transcription: Text transcription: frac{d x}{d t}+y=t^{2} -x+frac{d y}{d t}=1

Problem 15E In Problems 3–18, use the elimination method to find a general solution for the given linear system, where differentiation is with respect to t.

In Problems 3-18, use the elimination method to find a general solution for the given linear system, where differentiation is with respect to . \(\frac{d x}{d t}+x+\frac{d y}{d t}=e^{A t}\), \(2 x+\frac{d^{2} y}{d t^{2}}=0\) Equation transcription: Text transcription: frac{d x}{d t}+x+frac{d y}{d t}=e^{A t} 2 x+frac{d^{2} y}{d t^{2}}=0

Problem 19E In Problems 19–21, solve the given initial value problem.

In Problems 19-21, solve the given initial value problem. \(\frac{d x}{d t}=2 x+y-e^{2 t} ; x(0)=1\), \(\frac{d y}{d t}=x+2 y ; y(0)=1\) Equation transcription: Text transcription: \frac{d x}{d t}=2 x+y-e^{2 t} ; x(0)=1 \frac{d y}{d t}=x+2 y ; y(0)=1

In Problems 3-18, use the elimination method to find a general solution for the given linear system, where differentiation is with respect to . \(x^{\prime \prime}+5 x-4 y=0\), \(-x+y^{\prime \prime}+2 y=0\) Equation transcription: Text transcription: x^{prime prime}+5 x-4 y=0 -x+y^{prime prime}+2 y=0

Problem 21E In Problems 19–21, solve the given initial value problem.

Verify that the solution to the initial value problem \(x^{\prime}=5 x-3 y-2\) \(x(0)=2\) , \(y^{\prime}=4 x-3 y-1\) \(y(0)=0\) satisfies \(|x(t)|+|y(t)| \rightarrow+\infty\) as \(t \longrightarrow+\infty\). Equation transcription: Text transcription: x^{prime}=5 x-3 y-2 x(0)=2 y^{prime}=4 x-3 y-1 y(0)=0 |x(t)|+|y(t)| rightarrow+\infty t longrightarrow+infty

Problem 23E In Problems 23 and 24, show that the given linear system is degenerate. In attempting to solve the system, determine whether it has no solutions or infinitely many solutions.

In Problems , use the elimination method to find a general solution for the given system of three equations in the three unknown functions \(x(t), y(t), z(t)\) \(x^{\prime}=3 x+y-z\), \(y^{\prime}=x+2 y-z\), \(z^{\prime}=3 x+3 y-z\) Equation transcription: Text transcription: x(t), y(t), z(t) x^{prime}=3 x+y-z y^{prime}=x+2 y-z z^{prime}=3 x+3 y-z

In Problems , use the elimination method to find a general solution for the given system of three equations in the three unknown functions \(x(t), y(t), z(t)\) \(x^{\prime}=4 x-4 z\), \(y^{\prime}=4 y-2 z\) \(z^{\prime}=2 x-4 y+4 z\) Equation transcription: Text transcription: x(t), y(t), z(t) x^{prime}=4 x-4 z y^{prime}=4 y-2 z z^{prime}=2 x-4 y+4 z

Problem 24E In Problems 23 and 24, show that the given linear system is degenerate. In attempting to solve the system, determine whether it has no solutions or infinitely many solutions.

Problem 25E In Problems 25–28, use the elimination method to find a general solution for the given system of three equations in the three unknown functions x (t), y(t), z(t).

In Problems , use the elimination method to find a general solution for the given system of three equations in the three unknown functions \(x(t), y(t), z(t)\) \(x^{\prime}=4 x-4 z\), \(y^{\prime}=4 y-2 z\) \(z^{\prime}=2 x-4 y+4 z\) Equation transcription: Text transcription: x(t), y(t), z(t) x^{prime}=4 x-4 z y^{prime}=4 y-2 z z^{prime}=2 x-4 y+4 z

Two large tanks, each holding 100 L of liquid, are interconnected by pipes, with the liquid flowing from tank A into tank B at a rate of 3 L/min and from B into A at a rate of 1 L/min (see Figure 5.2). The liquid inside each tank is kept well stirred. A brine solution with a concentration of 0.2 kg/L of salt flows into tank A at a rate of 6 L/min. The (diluted) solution flows out of the system from tank A at 4 L/min and from tank B at 2 L/min. If, initially, tank A contains pure water and tank B contains 20 kg of salt, determine the mass of salt in each tank at time \(t \geq 0\). Equation transcription: Text transcription: t geq 0

Problem 32E In Problem 31, 3 L/min of liquid flowed from tank A into tank B and 1 L/min from B into A. Determine the mass of salt in each tank at time t ? 0 if, instead, 5 L/min flows from A into B and 3 L/min flows from B into A, with all other data the same. Problem: 31 -Two large tanks, each holding 100 L of liquid, are interconnected by pipes, with the liquid flowing from tank A into tank B at a rate of 3 L/min and from B into A at a rate of 1 L/min (see Figure 5.2). The liquid inside each tank is kept well stirred. A brine solution with a concentration of 0.2 kg/L of salt flows into tank A at a rate of 6 L/min. The (diluted) solution flows out of the system from tank A at 4 L/min and from tank B at 2 L/min. If, initially, tank A contains pure water and tank B contains 20 kg of salt, determine the mass of salt in each tank at time t ? 0.

In Problems 29 and 30, determine the range of values (if any) of the parameter that will ensure all solutions , of the given system remain bounded as \(\frac{d x}{d t}=\lambda x-y\) \(\frac{d y}{d t}=3 x+y\) Equation transcription: Text transcription: frac{d x}{d t}=lambda x-y frac{d y}{d t}=3 x+y

In Problems 29 and 30, determine the range of values (if any) of the parameter that will ensure all solutions , of the given system remain bounded as \(\frac{d x}{d t}=x+\lambda y\) \(\frac{d y}{d t}=x-y\) Equation transcription: Text transcription: frac{d x}{d t}=x+lambda y frac{d y}{d t}=x-y

Feedback System with Pooling Delay. Many physical and biological systems involve time delays. A pure time delay has its output the same as its input but shifted in time. A more common type of delay is pooling delay. An example of such a feedback system is shown in Figure on page 252 . Here the level of fluid in tank determines the rate at which fluid enters tank A. Suppose this rate is given by \(R_{1}(t)=\alpha\left[V-V_{2}(t)\right]\), where and are positive constants and \(V_{2}(t)\) is the volume of fluid in tank at time . (a) If the outflow rate \(R_{3}\) from tank is constant and the flow rate \(R_{2}\) from tank into is \(R_{2}(t)=K V_{1}(t)\), where is a positive constant and \(V_{1}(t)\) is the volume of fluid in tank at time , then show that this feedback system is governed by the system \(\frac{d v_{1}}{d t}=\alpha\left(V-V_{2}(t)\right)-K V_{1}(t)\) \(\frac{d v_{1}}{d t}=K V_{1}(t)-R_{3}\) (b) Find a general solution for the system in part (a) when \(\alpha=5(\min )^{-1}, V=20 L, K=2(\min )^{-1}\) and . (c) Using the general solution obtained in part (b), what can be said about the volume of fluid in each of the tanks as \(t \rightarrow+\infty\) ? Equation transcription: Text transcription: R{1}(t)=alpha[V-V{2}(t)] V{2}(t) R{3} R{2}(t)=K V{1}(t) R{2} V{1}(t) frac{d v{1}}{d t}=alpha(V-V{2}(t))-K V{1}(t) frac{d v{1}}{d t}=K V{1}(t)-R{3} alpha=5(min )^{-1}, V=20 L, K=2(min )^{-1} t rightarrow+infty

Problem 33E In Problem 31, assume that no solution flows out of the system from tank B, only 1 L/min flows from A into B, and only 4 L/min of brine flows into the system at tank A, other data being the same. Determine the mass of salt in each tank at time t ? 0. Problem: 31 -Two large tanks, each holding 100 L of liquid, are interconnected by pipes, with the liquid flowing from tank A into tank B at a rate of 3 L/min and from B into A at a rate of 1 L/min (see Figure 5.2). The liquid inside each tank is kept well stirred. A brine solution with a concentration of 0.2 kg/L of salt flows into tank A at a rate of 6 L/min. The (diluted) solution flows out of the system from tank A at 4 L/min and from tank B at 2 L/min. If, initially, tank A contains pure water and tank B contains 20 kg of salt, determine the mass of salt in each tank at time t ? 0.

In Problems 3–18, use the elimination method to find a general solution for the given linear system, where dif-ferentiation is with respect to \(t\). \(x^{\prime \prime}+y^{\prime \prime}-x^{\prime}=2 t\), \(x^{\prime \prime}+y^{\prime}-x+y=-1\) Equation Transcription: Text Transcription: t x''+y''-x'=2t x''+y'-x+y=-1