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### Course: AP®︎/College Calculus BC>Unit 7

Lesson 2: Verifying solutions for differential equations

# Verifying solutions to differential equations

We can check whether a potential solution to a differential equation is indeed a solution. What we need to do is differentiate and substitute both the solution and the derivative into the equation.

## Want to join the conversation?

• How to find a solution or set of solutions of differential equations?
• That's the main question of this entire field. All the videos and questions are here to answer it.
• What does it mean for f(x) to be a solution of f'(x)? Is there a graphical way to describe this?
• It would mean that taking the derivative of f(x) results int he same equation. The only functiont hat does that though is e^x

If you have other constraints though, like in f'(x)=f(x)-x there is more than just one function to satisfy the condition.

Maybe a better way of explaining it is saying if you have some f(x) and find f'(x) is there some other way to get it to its derivative other than taking it's derivative.

So let's say f(x) = 5x^2 so then f'(x)=10x. Now what can we do to 5x^2 without calculus to make it 10x? We could multiply it by 2/x. so we could say f'(x) = f(x)(2/x) and 5x^2 would be a solution to that.

So I guess you could say it's graph transformations to get to the graph of f'(x)
• Are units (meters, seconds, m/s) can be used in differential equations? Or is it just the number that can be used?
• It depends: in word problems it is often the case that the solution is looking for a rate (meters/sec, liters/sec, etc.). However, when faced with a problem such as y'' - 2y' + y = 0 the solution will be a function y = Ae^x + Bxe^x, where A & B are real values. No units, no measurements, just a good ol' fashioned function.
• I'm gonna fail😭😭😭 what am i learning
• Why does Sal use y=4x to get y=x^4, and how does he get y=4x to get y=x^4
• Why does Sal use y=4x to get y=x^4, and how does he get y=4x to get y=x^4?
• Why we find solution of differential equation ?
• Observation that I make (don't know if it's true or not):

if we are given 1st order differential equation: y' = y + cx^n , we can guess that the y is an unknown polynomial with degree of n
(1 vote)
• Yes, that's right.

For example with 𝑛 = 3,
we can let 𝑦 = 𝐴𝑥³ + 𝐵𝑥² + 𝐶𝑥 + 𝐷 ⇒ 𝑦' = 3𝐴𝑥² + 2𝐵𝑥 + 𝐶

𝑦' = 𝑦 + 𝑐𝑥³
⇒ 3𝐴𝑥² + 2𝐵𝑥 + 𝐶 = (𝐴 + 𝑐)𝑥³ + 𝐵𝑥² + 𝐶𝑥 + 𝐷

𝐴 = −𝑐
𝐵 = 3𝐴 = −3𝑐
𝐶 = 2𝐵 = −6𝑐
𝐷 = 𝐶 = −6𝑐

The leading coefficient is −𝑐, which then determines all the other coefficients.
This is of course not a formal proof for the general case, but I think you can imagine that this must be what happens for any whole number 𝑛.