Solving Differential Equations: Sinh & Cosh Maclaurin Series

In summary, the conversation discusses finding the Maclaurin series for sinh(x) and cosh(x) using the Maclaurin series for ex and the definitions of sinh(x) and cosh(x). It also mentions computing the radius of convergence for each series. The first part of the conversation involves showing that the function p(t) satisfies a specific differential equation. The second part asks what happens to p(t) as t > (A^(-c)/c) from the left. The conversation ends with a comment about the suspicion of this being a homework question.
  • #1
Abelian_Math
3
0
1.
Let c be a positive number, and let A > 0 represent the initial value of a population.
a) Show that the function p(t) = (A^(-c) - ct)^(-1/c) satisfies the differential equation
p'(t) = (p(t))^(1+c)
b) What happens to p(t) as t > (A^(-c)/c) from the left?


2. Find the Maclaurin series for the functions sinh(x) and cosh(x) by using the Maclaurin
series for ex and the defnitions of sinh(x) and cosh(x) in terms of ex. Compute the radius
of convergence for each series.
 
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  • #2
Have you not even tried to do this yourself? And this looks suspiciously like homework.
 

FAQ: Solving Differential Equations: Sinh & Cosh Maclaurin Series

1. What are differential equations?

Differential equations are mathematical equations that describe how a quantity changes over time or space. They involve derivatives, which represent how the quantity changes at a particular point.

2. What is the purpose of solving differential equations?

The purpose of solving differential equations is to find a function or set of functions that satisfy the equation and accurately model the behavior of a system. This allows us to make predictions and understand the underlying dynamics of the system.

3. What are sinh and cosh Maclaurin series?

Sinh and cosh Maclaurin series are infinite series that represent the hyperbolic sine and cosine functions, respectively, as an infinite sum of terms involving powers of the input. These series can be used to approximate the values of these functions at any point.

4. How are sinh and cosh Maclaurin series used in solving differential equations?

Since differential equations often involve trigonometric or hyperbolic functions, we can use sinh and cosh Maclaurin series to simplify these functions and make them easier to work with. This allows us to solve the differential equation and find the desired solution.

5. What are some real-world applications of solving differential equations using sinh and cosh Maclaurin series?

Differential equations and Maclaurin series have a wide range of applications in fields such as physics, engineering, and economics. For example, they can be used to model the motion of a pendulum, the growth of a population, or the flow of electricity in a circuit.

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