# Math Insight

### Introduction to simple linear differential equations

Math 1241, Fall 2020
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Due date: Nov. 20, 2020, 11:59 p.m.
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Total points: 3
1. To begin, let's review the derivative of the exponential function.
1. If $f(t)=e^{2 t}$, what is $\diff{f}{t}$?
If $g(t)=e^{- 5 t}$, what is $\diff{g}{t}$?
If $h(t) = e^{a t}$ for a constant $a$, what is $\diff{h}{t}$?
2. Let's say that the function $u(t)$ is one of those three functions. It is either $u(t)=e^{2 t}$, $u(t)=e^{- 5 t}$ or $u(t)=e^{a t}$. To determine the identity of $u(t)$, you are given the information that the derivative $\diff{u}{t}$ is the same thing as two times the function itself, i.e., that $\diff{u}{t} = 2 u(t)$. What is $u(t)$? $u(t)=$
3. Let's say that the function $s(t)$ is also one of those three functions. You are given the additional information that the derivative $\diff{s}{t}$ is the same thing as $-5$ times the function itself, i.e., that $\diff{s}{t} = -5 s(t)$. What is $s(t)$? $s(t) =$
4. Equations like $\diff{u}{t} = 2u(t)$ are called differential equations. We say that $u(t)$ is a solution to that differential equation. The above function $h(t)=e^{a t}$ is a solution to a similar differential equation. What is the differential equation that $h(t)$ solves?

$\displaystyle \diff{h}{t} =$

2. Consider the functions $f(t)=3 e^{7 t}$, $g(t)=- 2 e^{7 t}$, and $h(t)=7 e^{7 t}$.
1. Calculate the derivatives of $f$, $g$, and $h$.

$\diff{f}{t} =$

$\diff{g}{t} =$

$\diff{h}{t} =$

2. The expressions for those derivatives should be equal to the same thing as the function itself, multiplied by a number.

$\diff{f}{t} =$
$\times f(t)$
$\diff{g}{t} =$
$\times g(t)$
$\diff{h}{t} =$
$\times h(t)$

3. The function $v(t)$ is one of those three function and is the solution to the differential equation $\diff{v}{t} = 7v(t)$. Can you determine whether $v(t)=3 e^{7 t}$, $v(t)=- 2 e^{7 t}$, or $v(t)=7 e^{7 t}$?
Which of these are possibilities for the solution $v(t)$?
4. If you were given the additional piece of information that $v(0)=3$, could you determine the identity of $v(t)$?
If so, what is $v(t)$? $v(t)=$
5. The function $w(t)$ is one of those three functions and is the solution to the differential equation $\diff{w}{t} = 7w$ with initial condition $w(0)=-2$. What is $w(t)$? $w(t)=$
6. Actually, we can solve this differential equation even if the answer is not one of those options. If $z(t)$ is a function that satisfies the differential equation $\diff{z}{t}=7z$ with initial condition $z(0)=11$, what is $z(t)$? $z(t)=$

3. Solve the following differential equations.
1. $\diff{u}{t} = 3 u{\left (t \right )}$
$u(0) = 5$

$u(t)=$

2. $\diff{v}{t} = - 4 v$
$v(0) = 7$

$v(t)=$

3. $s'(t) = s$
$s(0) = -2$

$s(t)=$

4. $z'(t) = - z{\left (t \right )}$
$z(0) = -5$

$z(t)=$

4. If we aren't given the initial conditions, we cannot determine the exact solution of the differential equation. For the following differential equations, give the general solution, which will be a function with an undetermined constant $C$.
1. $\diff{y}{t}=2 y$

$y(t)=$

2. $x'(t)=- 3 x{\left (t \right )}$

$x(t)=$

3. $\diff{m}{t}=11 m{\left (t \right )}$.

$m(t)=$

4. $v'(t)=v$.

$v(t)=$