Math / Calculus

QuestionGiven $\frac{d y}{d t}=7 y$ and $y(0)=450$ , determine $y(t)$ $y(t)=$ $\square$

Studdy Solution

STEP 1

What is this asking? We're given a differential equation that tells us how a quantity $y$ changes over time $t$ , and we know its initial value.
We need to find a formula for $y$ at any time $t$ . Watch out! Remember that the solution to this type of differential equation involves an exponential function, not a linear one!

STEP 2

1. Solve the differential equation.
2. Apply the initial condition.

STEP 3

We start with our differential equation: $\frac{dy}{dt} = 7y$ To solve this, we want to separate the variables $y$ and $t$ .
We can do this by dividing both sides by $y$ and multiplying both sides by $dt$ : $\frac{1}{y} \cdot \frac{dy}{dt} \cdot dt= 7 \cdot dt$ $\frac{1}{y} dy = 7 dt$ Now, our variables are nicely separated!

STEP 4

Next, we integrate both sides.
Don't forget the constant of integration, which we'll call $C$ , on the $t$ side: $\int \frac{1}{y} dy = \int 7 dt$ $\ln|y| = 7t + C$

STEP 5

To solve for $y$ , we'll exponentiate both sides using $e$ : $e^{\ln|y|} = e^{7t + C}$ $|y| = e^{7t} \cdot e^C$ Since $e^C$ is just another constant, we can rewrite it as $A$ : $|y| = Ae^{7t}$ We can drop the absolute value if we allow $A$ to be either positive or negative: $y = Ae^{7t}$ Great! We've found the general solution to the differential equation!

STEP 6

We're given that $y(0) = 450$ .
This means when $t = 0$ , $y = 450$ .
Let's substitute these values into our general solution: $450 = Ae^{7 \cdot 0}$ $450 = Ae^0$ Since any number raised to the power of zero is one, we have: $450 = A \cdot 1$ $450 = A$ So, $A = \mathbf{450}$ !

STEP 7

Now, we substitute our value of $A$ back into the general solution: $y = 450e^{7t}$ This is the particular solution to our differential equation, taking into account the initial condition!

STEP 8

$y(t) = 450e^{7t}$

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms