Homework Problems for Section 6.4 (ODEs with discontinuous forcing functions)
  1. Problem. Find the solution of the initial value problem.

  2. (1) $y'' + 3y' +2y = f(t);\qquad y(0)=0, \quad y'(0)=0; \qquad f(t)=\begin{cases}2,\qquad 0\leq t <10\\ 0,\qquad t\geq 10\end{cases}$
  3. (2) $y'' - 4y' +3y = 6u_2(t); \qquad y(0)=0, \quad y'(0)=2$
  4. (3) $y'' + 4y = 4u_{\pi}(t) - 4u_{3\pi}(t); \qquad y(0)= 0, \quad y'(0)=0$
  5. (4) $y'' - 4y' + 4y = f(t); \qquad y(0)= 0, \quad y'(0)=1; \qquad f(t)=\begin{cases}-4,\qquad 0\leq t <1\\ 0,\qquad t\geq 1\end{cases}$
  6. Answers
    (1) $y=[1+e^{-2t}-2e^{-t}]-u_{10}(t)[1+e^{-2(t-10)}-2e^{-(t-10)}]$
    (2) $y=e^{3t}-e^{t}+u_2(t)[2-3e^{t-3}+e^{3(t-3)}]$
    (3) $y=u_{\pi}(t)[1-\cos(2t-2\pi)]-u_{3\pi}(t)[1-\cos(2t-6\pi)]$ (or $y=u_{\pi}(t)(1-\cos(2t))-u_{3\pi}(t)(1-\cos(2t))$)
    (4) $y=-1+e^{2t}-te^{2t}+ u_1(t)[1-e^{2(t-1)}+2(t-1)e^{2(t-1)}]$
Homework Problems for Section 3.5 (Nonhomogeneous equations)
  1. Problem. Find the general solution of the given nonhomogeneous equation. The unknown $y$ is a function of the independent variable $t$.
  2. (1) $y'' -2y' -3y=3e^{2t}$
  3. (2) $y'' + 2y' + 5y=3\sin(2t)$
  4. (3) $y''-4y'+4y = -7t+4t^2$
  5. (4) $y''+y'-6y = 12e^{3t}+12e^{-2t}$
  6. (5) $y''+y'-6y = 2e^{2t}$
  7. (6) $y''+2y' = 4t$
  8. (7) $y''-3y'-4y = 5e^{-t}$
  9. (8) $y''+4y = 4\sin(2t)$

  10. Answers
    (1) $y=c_1e^{3t}+c_2e^{-t}-e^{2t}$
    (2) $y=c_1e^{-t}\cos(2t)+c_2e^{-t}\sin(2t)+\frac{3}{17}\sin(2t)-\frac{12}{17}\cos(2t)$
    (3) $y = c_1e^{2t}+c_2te^{2t}+t^2+\frac{t}{4}-\frac{1}{4}$
    (4) $y = c_1e^{-3t}+c_2e^{2t}+2e^{3t}-3e^{-2t}$
    (5) $y = c_1e^{-3t}+c_2e^{2t}+\frac{2}{5}te^{2t}$
    (6) $y = c_1+c_2e^{-2t}+t^2-t$
    (7) $y = c_1e^{4t}+c_2e^{-t}-te^{-t}$
    (8) $y = c_1\cos(2t)+c_2\sin(2t)-t\cos(2t)$
Homework Problems for Section 2.4 (Bernoulli equations)
  1. Problem. Find the general solution of the given Bernoulli equation in explicit form. The unkown $y$ is a function of the independent variable $t$.
  2. (a) $y' = 2y +e^{t}y^3$
  3. (b) $y' = 3y-y^2$
  4. (c) $t^2y' - 2ty -y^3=0$
  5. Answers
    (a) $\displaystyle y=\pm\sqrt{\dfrac{1}{-\frac{2}{5}e^{t}+c\,e^{-4t}}}$;
    (b) $y=\dfrac{1}{\frac{1}{3} + c\,e^{-3t}}$;
    (c) $\displaystyle y=\pm\sqrt{\dfrac{1}{-\frac{2}{3t}+\frac{c}{t^4}}}$
Homework Problems for Section 2.3 (Applications of ODEs)
  1. Problem 1. A tank initially contains $100$ gallons of salt water with a concentration of $2$ lbs/gal. Salt water containing $0.25$ lbs/gal of salt is entering the tank at a rate of $4$ gal/min and at the same time, the well-stirred mixture is draining from the tank at the same rate. Find the amount of salt $Q(t)$ in the tank at any time $t$.
  2. Problem 2. A tank with a large capacity originally contains $200$ gal of water with $100$ lbs of salt in solution. Water containing $1$ lb of salt per gallon flows into the tank at a rate of $4$ gal/min and the mixture is allowed to flow out at a rate of $2$ gal/min. Find the amount of salt $Q(t)$ in the tank at any time $t$.
  3. Problem 3. A tank originally contains $500$ L of a certain chemical solution with a concentration of $2$ g/L. The same type of chemical solution with concentration of $3$ g/L flows into the tank at a rate of $1$ L/min and the mixture is allowed to flow out at a rate of $2$ L/min. Find the amount of salt $Q(t)$ in the tank at any time $t$, before the tank is drained out.
  4. Answers
    (1) $Q(t) = 25+ 175e^{-t/25}$;
    (2) $Q(t) = 2(t+100) - \frac{10,000}{t+100}$; or equivalently, $Q(t) = \frac{2t^2+400t+10,000}{t+100}$;
    (3) $Q(t) = 3(500-t)-\frac{(500-t)^2}{500}$; or equivalently, $Q(t) = 1000 - t - \frac{t^2}{500}$
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