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Solve the following differential equations. Use your calculator to draw a family of solutions. Are there certain initial conditions that change the behavior of the solution?$$y^{\prime}=3 e^{t / 3}-2 y$$

$y(t)=\frac{9}{7}e^\frac{t}{3}+Ce^{-2t} $The solutions change their behaviors on using $y(0)$ approaches $\frac{3}{2}$

Calculus 2 / BC

Chapter 4

Introduction to Differential Equations

Section 5

First-order Linear Equations

Differential Equations

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Section four, not five Problem to 34 here were asked to solve an ordinary first order linear differential equation and then look at the graph of the family of solutions. So to put this in standard form, this is going to be y prime plus two. Why is equipped with three e to the t over three and then are integrating factor? It's going to be e to the integral two d x. So this is e to the to t. So this problem becomes why e to the to t prime is equal to three e to the t over three either the to t and so this is equal to three and then e toothy 7/3 t. So now the solution for this guy will be to integrate both sides of this equation. And so you end up with why he to the to t is equal to. So when you integrate this, you're going to have a three he to the 7/3 t and then you divide by 7/3 which is 3/7 plus a constant of integration. So this gives us why he to the to t is equal to 9/7. e to the 7/3 t plus c And then now to finally solve this for why you're gonna have why is equipped on 9/7 e to the 7/3 t minus two t plus C e to the minus two t So 7/3 money, 6/3. So this is why people 9/7 e to the t over three plus ce to the miners to t So that is our family of solutions for this. So if you take a look at the graph here, this is what we see. What does it look like for different values of C? So one of the always good things to do is to look at when CIA zero, when c is zero. He only had the first term. So, hoops, let me get it to be zero when c is zero, you only have that first term and so 9/7 e to the exercise. You see that one exponential every other time you've got the sum of two exponential functions, and so you see the various different behaviors for constants of integration and you see a change there as well. Um, where it starts to go up so These are the family of solutions we could look at that is it animates. But these are the family of solutions for this particular, um, or new to natural question.

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