Hi all.
I need help understanding something my book dosn't go over well. For example dy/dx=-4y is a given solution and I need to find the differential equation. I simply isolate y then integrate. Getting y=e^(-4x) which is correct. My question is for something like (d^2y)/(dx^2)=-16y how is this process different because I cant get to the answer by just doing the integral twice. The answer by the way is y=sin(4x). Please help explain this Im taking an online class and the book dosn't explain this?
thank!
You have
(D^2+16)y = 0
so the solutions are
e^(±4ix) or sin and cos
(sin 4x)" = -16sin(4x)
(cos 4x)" = -16cos(4x)
If the answer is just sin(4x) then there must be some other conditions
I cant really follow how you got your work...
Hello!
The process of finding the differential equation given a solution is called "reverse engineering" or "going backward." In general, if you are given a solution y = f(x), you need to find a differential equation that satisfies this solution.
Let's go through the steps for finding the differential equation for your given solution, y = sin(4x):
Step 1: Find the derivative of y with respect to x.
dy/dx = 4cos(4x)
Step 2: Find the second derivative of y with respect to x.
(d^2y)/(dx^2) = -16sin(4x)
Here, you can see that the differential equation associated with y = sin(4x) is (d^2y)/(dx^2) = -16y.
However, it's important to note that finding the differential equation from the solution is not a straightforward process. In many cases, it involves recognizing patterns or applying mathematical techniques specific to the problem.
In the case of dy/dx = -4y, the equation is separable, and you can integrate it directly to find the solution y = e^(-4x). However, finding the differential equation for (d^2y)/(dx^2) = -16y requires knowledge of the relationship between trigonometric functions and their derivatives.
So, there isn't a general method to find the differential equation from a given solution. It often requires familiarity with the particular area of mathematics and different techniques suitable for different types of functions.
I hope this explanation clarifies things for you. Let me know if you have any further questions!