Show any function satisfies the differential equation

[MarkSA]

Hello,

I'm studying a short section in calculus 2 on hyperbolic functions. Suddenly I come across this question:

Show that any function of the form:
y = A*sinh(mx) + B * cosh(mx)
satisfies the differential equation: y'' = (m^2)y

I feel like a deer caught in the headlights. Differential equation? How do I do this? The y double prime looks innocent enough but I don't have a clue what this question is asking me to do.

 

[masi]

Hello,

I'm studying a short section in calculus 2 on hyperbolic functions. Suddenly I come across this question:

Show that any function of the form:
y = A*sinh(mx) + B * cosh(mx)
satisfies the differential equation: y'' = (m^2)y

I feel like a deer caught in the headlights. Differential equation? How do I do this? The y double prime looks innocent enough but I don't have a clue what this question is asking me to do.

A differential equation is simply an equation that involves some function (in this case y) and its derivatives. Typically with differential equations, you are given the equation and must find the function that satisfies it. For example, if we're given \(\displaystyle y' = 2x\), then it's clear that the solution is of the form \(\displaystyle y = x^2 + C\). In your case, you're given a function and you must show it satisfies the equation. All this amounts to in your case is finding \(\displaystyle y''\) and showing it is equal to \(\displaystyle m^2 y.\). A really short example that is essentially what you have to do, but with different functions: Suppose \(\displaystyle f(x) = e^{Cx}\). Show that f satisfies \(\displaystyle f^{(n)}(x) = C^n f(x)\) for all natural numbers n. Well we can use the chain rule (and a bit of induction) to show that the nth derivative of \(\displaystyle e^{Cx}\) is \(\displaystyle C^n e^{Cx}\), thus it satisfies the equation.

Hopefully this was helpful.

 

[Dr. Flim-Flam]

y =f(x ). Now find y ', then y " and you are done.