Is $$y=3e^{2x}$$ a solution to $$\frac{dy}{dx}=2y$$ ?

Yes. $$\frac{d}{dx}\left(3e^{2x}\right)=6e^{2x}$$ and $$2y=6e^{2x}$$ You must check LHS = RHS of Diff Eq.

Strategy for sketching a slope field?

Make a table of slopes by plugging in several (x, y) values into $$\frac{dy}{dx}$$ .

Common mistake with slope fields...

Forgetting that the slope depends on both x and y — not just one variable.

What can you infer from a slope field?

General shape of solutions, where they increase or decrease, and behavior near initial conditions.

How do you sketch a solution curve using a slope field?

Start at the initial point and follow the direction of the tiny slope segments.

What is Euler’s Method used for?

Approximating solutions to Diff Eqs when an exact solution is hard or impossible.

$$y-y_1=\frac{dy}{dx}\left(x-x_1\right)$$

What does each step in Euler's Method represent?

A linear approximation using the current slope.

What is the goal of separation of variables?

Rewrite the equation so all y-terms are on one side and x-terms on the other before integrating.

Example: Solve $$\frac{dy}{dx}=3y$$

Separate → $$\frac{1}{y}dy=3\ dx$$ Integrate → $$\ln\left|y\right|=3x+C$$ Solve for y → $$y=\pm Ce^{3x}$$ Usually have to figure if + or - is the correct sign based on additional information.

How do you find a particular solution?

First find the general solution, then plug in the initial condition to solve for C.

General form for exponential growth/decay?

$$\frac{dy}{dx}=ky$$ , solution: $$y=Ce^{kt}$$ k > 0 means growth k < 0 means decay

What does the logistic model account for?

Limited resources and a carrying capacity L.

General logistic Diff Eq?

$$\frac{dy}{dt}=ky\left(1-\frac{y}{L}\right)$$ or $$\frac{dy}{dt}=\frac{k}{L}\cdot y\cdot\left(L-y\right)$$

Behavior of logistic curve?

S-shaped: starts growing exponentially, slows down near L. Fastest growth rate is at L/2 The curve has an inflection at L/2

How do you identify a telescoping series?

A series is telescoping when many of its terms cancel out in the partial sum. Write out 3-5 terms to spot the pattern.

Classic clues for telescoping series?

The general term is a difference of two expressions, like: $$a_n=f\left(n\right)-f\left(n-1\right)$$ or $$\frac{1}{n\left(n+1\right)}=\frac{1}{n}-\frac{1}{n+1}$$

What happens when you write out the first few terms of telescoping series?

You start to see the cancellation happen: $$\left(b_1-b_0\right)+\left(b_2-b_1\right)+\left(b_3-b_2\right)+...+\left(b_n-b_{\left(n-1\right)}\right)=b_n-b_0$$

What are the convergence criteria for telescoping series?

The series converges if and only if $$\Sigma\left(b_n\right)$$ is finite

Why are partial sums of telescoping series easy to compute?

Σ (b n - b n-1 ) = b n - b 0 and $$\Sigma\left(\frac{1}{n\left(n+1\right)}\right)=\lim_{N\rightarrow\infty}\left(1-\frac{1}{N+1}\right)=1$$

Unit 7 & 10 Flashcards

Student preview

27 questions

Show all answers

1.

FLASHCARD QUESTION

Front

What is a differential equation?

Back

An equation involving a function and its derivatives, often describing rates of change.

2.

FLASHCARD QUESTION

Front

How do differential equations model real-world problems?

Back

They represent how quantities change over time, like population growth or radioactive decay.

3.

FLASHCARD QUESTION

Front

Common example of a model?

Back

4.

FLASHCARD QUESTION

Front

Back

5.

FLASHCARD QUESTION

Front

How do you verify a differential equation solution?

Back

Differentiate the given function, plug into the original differential equation, and check if it holds true.

6.

FLASHCARD QUESTION

Front

Back

7.

FLASHCARD QUESTION

Front

What does each segment in a slope field represent?

Back

The slope of a solution curve at that point (x, y), based on the Diff Eq.

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