In each of Problems 26 through 33 draw a direction field for the given differential equation. Based on the direction field, determine the behavior of $y$ as $t\to \infty$. If this behavior depends on the initial value of $y$ at $t = 0$, describe this dependency. Note that the right sides of these equations depend on t as well as y; therefore their solutions can exhibit more complicated behavior than those in the text.

28) $y'=e^{-t}+y$

31) $y'=2t-1-y^2$

5) **Undetermined Coefficients.** Here is an alternative way to solve the equation
$$dy/dt = ay-b.\tag{i}$$

(a) Solve the simpler equation $$dy/dt = ay \tag{ii}$$ Call the solution $y_1(t)$.

(b) Observe that the only difference between Eqs. (i) and (ii) is the constant $-b$ in Eq. (i). Therefore it may seem reasonable to assume that the solutions of these two equations also differ only by a constant. Test this assumption by trying to find a constant $k$ such that $y = y_1(t) + k$ is a solution of Eq. (i).

(c) Compare your solution from part (b) with the solution given in the text in Eq. (17).

Note: This method can also be used in some cases in which the constant $b$ is replaced by a function $g(t)$. It depends on whether you can guess the general form that the solution is likely to take. This method is described in detail in Section 3.5 in connection with second order equations.

8) Consider a population $p$ of field mice that grows at a rate proportional to the current population, so that $dp/dt = rp$.

Find the rate constant r if the population doubles in 30 days.

Find $r$ if the population doubles in $N$ days

In each of Problems 1 through 6 determine the order of the given differential equation; also state whether the equation is linear or nonlinear.

\begin{align*} &1)\ t^2\frac{d^2y}{dt^2}+t\frac{dy}{dt}+2y=\sin(t) &&2)\ (1+y^2)\frac{d^2y}{dt^2} +t\frac{dy}{dt}+y=e^t\\ &3)\ \frac{d^4y}{dt^4}+\frac{d^3y}{dt^3}+\frac{d^2y}{dt^2}+\frac{dy}{dt}+y=1 &&4)\ \frac{dy}{dt}+ty^2=0\\ &5) \frac{d^2y}{dt^2}+\sin(t+y)=\sin(t) &&6)\ \frac{d^3y}{dt^3}+t\frac{dy}{dt}+\cos^2(t)\cdot y=t^3 \end{align*}

In each of Problems 7 through 14 verify that each given function is a solution of the differential equation.

7) $y''-y=0$; $y_1(t)=e^t$, $y_2(t)=\cosh(t)$

8) $y''+2y'-3y=0$; $y_1(t)=e^{-3t}$, $y_2(t)=e^t$

10) $y'''' + 4y''' + 3y = t$; $y_1(t) = t/3$, $y_2(t) = e^{-t} + t/3$

11) $2t^2y'' + 3ty' - y = 0,\ t > 0$; $y_1(t) = t^{1/2}$, $y_2(t) = t^{-1}$

13) $y'' + y = \sec (t),\ 0< t < \pi/2$; $y = \cos(t) \ln (\cos (t)) + t \sin (t)$

14) $y'- 2ty = 1$; $y = e^{t^2}\int_0^t e^{-s^2}\,ds+e^{t^2}$

In each of Problems 15 through 18 determine the values of $r$ for which the given differential equation has solutions of the form $y = e^{rt}$.

17) $y'' + y' - 6y = 0$

18) $y''' - 3y'' + 2y' = 0$

In each of Problems 21 through 24 determine the order of the given partial differential equation; also state whether the equation is linear or nonlinear. Partial derivatives are denoted by subscripts.

21) $u_{xx} + u_{yy} + u_{zz} = 0$

22) $u_{xx} + u_{yy} + uu_x + uu_y + u = 0$

23) $u_{xxxx} + 2u_{xxyy} + u_{yyyy} = 0$

24) $u_t + uu_x = 1 + u_{xx}$