Projectile Motion with Air Resistance Determine a system

In Problems 1–20 solve the given system of differential equations by systematic elimination. \(\frac{d x}{d t}=2 x-y\) \(\frac{d y}{d t}=x\) Text Transcription: frac{d x}{d t}=2 x-y frac{d y}{d t}=x

In Problems 1–20 solve the given system of differential equations by systematic elimination. \(\frac{d x}{d t}=4 x+7 y\) \(\frac{d y}{d t}=x-2 y\) Text Transcription: frac{d x}{d t}=4 x+7 y frac{d y}{d t}=x-2 y

(a) Reread Problem 8 of Exercises 3.3. In that problem you were asked to show that the system of differential equations \(\frac{d x_{1}}{d t}=-\frac{1}{50} x_{1}\) \(\frac{d x_{2}}{d t}=\frac{1}{50} x_{1}-\frac{2}{75} x_{2}\) \(\frac{d x_{3}}{d t}=\frac{2}{75} x_{2}-\frac{1}{25} x_{3}\) is a model for the amounts of salt in the connected mixing tanks A, B, and C shown in Figure 3.3.7. Solve the system subject to \(x_{1}(0)=15\), \(x_{2}(t)=10\), \(x_{3}(t)=5\). (b) Use a CAS to graph \(x_{1}(t)\), \(x_{2}(t)\), and \(x_{3}(t)\) in the same coordinate plane (as in Figure 4.9.1) on the interval [0, 200]. (c) Because only pure water is pumped into Tank A, it stands to reason that the salt will eventually be flushed out of all three tanks. Use a root-finding application of a CAS to determine the time when the amount of salt in each tank is less than or equal to 0.5 pound. When will the amounts of salt \(x_{1}(t)\), \(x_{2}(t)\), and \(x_{3}(t)\) be simultaneously less than or equal to 0.5 pound? Text Transcription: fracd x_1dt=-frac150x_1 fracd x_2dt=\frac150x_1-frac275x_2 fracd x_3dt=\frac275x_2-frac125x_3 x_1(0)=15 x_2(t)=10 x_3(t)=5 x_1(t) x_2(t) x_3(t)

In Problems 1–20 solve the given system of differential equations by systematic elimination. \(\frac{d x}{d t}=-y+t\) \(\frac{d y}{d t}=x-t\) Text Transcription: frac{d x}{d t}=-y+t frac{d y}{d t}=x-t

In Problems 1–20 solve the given system of differential equations by systematic elimination. \(\frac{d x}{d t}-4 y=1\) \(\frac{d y}{d t}+x=2\) Text Transcription: frac{d x}{d t}-4 y=1 frac{d y}{d t}+x=2

In Problems 1–20 solve the given system of differential equations by systematic elimination. \(\left(D^{2}+5\right) x-\quad 2 y=0\) \(-2 x+\left(D^{2}+2\right) y=0\) Text Transcription: left(D^{2}+5\right) x-quad 2 y=0 -2 x+\left(D^{2}+2\right) y=0

In Problems 1–20 solve the given system of differential equations by systematic elimination. \((D+1) x+(D-1) y=2\) \(3x + (D + 2)y =- 1\) Text Transcription: (D+1) x+(D-1) y=2 -2 x+\left(D^{2}+2\right) y=0

In Problems 1–20 solve the given system of differential equations by systematic elimination. \(\frac{d^{2} x}{d t^{2}}=4 y+e^{t}\) \(\frac{d^{2} y}{d t^{2}}=4 x-e^{t}\) Text Transcription: frac{d^{2} x}{d t^{2}}=4 y+e^{t} frac{d^{2} y}{d t^{2}}=4 x-e^{t}

In Problems 1–20 solve the given system of differential equations by systematic elimination. \(\frac{d^{2} x}{d t^{2}}+\frac{d y}{d t}=-5 x\) \(\frac{d x}{d t}+\frac{d y}{d t}=-x+4 y\) Text Transcription: frac{d^{2} x}{d t^{2}}+\frac{d y}{d t}=-5 x frac{d x}{d t}+\frac{d y}{d t}=-x+4 y

In Problems 1–20 solve the given system of differential equations by systematic elimination. \(D x+\quad D^{2} y=e^{3 t}\) \((D+1) x+(D-1) y=4 e^{3 t}\) Text Transcription: D x+\quad D^{2} y=e^{3 t} (D+1) x+(D-1) y=4 e^{3 t}

In Problems 1–20 solve the given system of differential equations by systematic elimination. \(\left(D^{2}-1\right) x-y=0\) \((D - 1)x + Dy = 0\) Text Transcription: left(D^{2}-1\right) x-y=0

In Problems 1–20 solve the given system of differential equations by systematic elimination. \(D^{2} x-\quad D y=t\) \((D+3) x+(D+3) y=2\) Text Transcription: D^{2} x-\quad D y=t (D+3) x+(D+3) y=2

In Problems 1–20 solve the given system of differential equations by systematic elimination. \(\left(2 D^{2}-D-1\right) x-(2 D+1) y=1\) \((D - 1)x + Dy = -1\) Text Transcription: left(2 D^{2}-D-1\right) x-(2 D+1) y=1

In Problems 1–20 solve the given system of differential equations by systematic elimination. \(2 \frac{d x}{d t}-5 x+\frac{d y}{d t}=e^{t}\) \(\frac{d x}{d t}-x+\frac{d y}{d t}=5 e^{t}\) Text Transcription: (2 \frac{d x}{d t}-5 x+\frac{d y}{d t}=e^{t} frac{d x}{d t}-x+\frac{d y}{d t}=5 e^{t}

In Problems 1–20 solve the given system of differential equations by systematic elimination. \(\frac{d x}{d t}+\frac{d y}{d t}=e^{t}\) \(-\frac{d^{2} x}{d t^{2}}+\frac{d x}{d t}+x+y=0\) Text Transcription: frac{d x}{d t}+\frac{d y}{d t}=e^{t} -frac{d^{2} x}{d t^{2}}+\frac{d x}{d t}+x+y=0

In Problems 1–20 solve the given system of differential equations by systematic elimination. \((D-1) x+\left(D^{2}+1\right) y=1\) \(\left(D^{2}-1\right) x+(D+1) y=2\) Text Transcription: (D-1) x+\left(D^{2}+1\right) y=1 left(D^{2}-1\right) x+(D+1) y=2

In Problems 1–20 solve the given system of differential equations by systematic elimination. \(D^{2} x-2\left(D^{2}+D\right) y=\sin t\) \(x + Dy = 0\) Text Transcription: D^{2} x-2\left(D^{2}+D\right) y=\sin t

In Problems 1–20 solve the given system of differential equations by systematic elimination. Dx = y Dy = z Dz = x

In Problems 1–20 solve the given system of differential equations by systematic elimination. \(D x+\quad z=e^{t}\) (D - 1)x + Dy + Dz = 0 \(x+2 y+D z=e^{t}\) Text Transcription: D x+\quad z=e^{t} x+2 y+D z=e^{t}

In Problems 1–20 solve the given system of differential equations by systematic elimination. \(\frac{d x}{d t}=6 y\) \(\frac{d y}{d t}=x+z\) \(\frac{d z}{d t}=x+y\) Text Transcription: frac{d x}{d t}=6 y frac{d y}{d t}=x+z frac{d z}{d t}=x+y

In Problems 1–20 solve the given system of differential equations by systematic elimination. \(\frac{d x}{d t}=-x+z\) \(\frac{d y}{d t}=-y+z\) \(\frac{d z}{d t}=-x+y\) Text Transcription: frac{d x}{d t}=-x+z frac{d y}{d t}=-y+z frac{d z}{d t}=-x+y

In Problems 21 and 22 solve the given initial-value problem. \(\frac{d x}{d t}=-5 x-y\) \(\frac{d y}{d t}=4 x-y\) \(x(1) = 0, y(1) = 1\) Text Transcription: frac{d x}{d t}=-5 x-y frac{d y}{d t}=4 x-y

In Problems 21 and 22 solve the given initial-value problem. \(\frac{d x}{d t}=y-1\) \(\frac{d y}{d t}=-3 x+2 y\) \(x(0) = 0, y(0) = 0\) Text Transcription: frac{d x}{d t}=y-1 frac{d y}{d t}=-3 x+2 y

Projectile Motion A projectile shot from a gun has weight w = mg and velocity v tangent to its path of motion. Ignoring air resistance and all other forces acting on the projectile except its weight, determine a system of differential equations that describes its path of motion. See Figure 4.9.2. Solve the system. [Hint: Use Newton’s second law of motion in the x and y directions.]

Projectile Motion with Air Resistance Determine a system of differential equations that describes the path of motion in Problem 23 if air resistance is a retarding force k (of magnitude k) acting tangent to the path of the projectile but opposite to its motion. See Figure 4.9.3. Solve the system. [Hint: k is a multiple of velocity, say, \(\beta \mathbf{v}\)]

Examine and discuss the following system: \(\begin{aligned}D x-2 D y &=t^{2} \\ (D+1) x-2(D+1) y &=1\end{aligned}\) Text Transcription: D x-2Dy&=t^2(D+1)x-2(D+1)y&=1

Reexamine Figure 4.9.1 in Example 3. Then use a root-finding application to determine when tank B contains more salt than tank A.