Help With Solving This Differential Equation

In summary, the given conversation discusses the slope of a function at any point (x,y) being 2e^x/(e^x+2), with the point (0,2ln3) being on the graph of f. The tangent line to the graph of f at x=0 is found to be y=2/3 x+2ln3. Using this tangent line, the approximate value of f(0.3) is found to be 2.397. The differential equation dy/(dx)=(2e^x)/(e^x+2) is solved with the initial condition f(0)=2ln3, but the integral ∫〖(2e^x)/(e^x+
  • #1
helpmeimdumb
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Homework Statement



1. The slope of a function at any point (x,y) is 〖2e〗^x/(e^x+2) . The point (0,2ln3) is on the graph of f.

(A) Write an equation of the tangent line to the graph of f at x=0.
(B) Use the tangent line in part A to approximate f(0.3) to the nearest thousandth.
(C) Solve the differential equation dy/(dx )=(2e^x)/(e^x+2) with the initial condition f(0)=2ln3.
(D) Use the solution in part C to find f(0.3) to the nearest thousandth.



Homework Equations





The Attempt at a Solution




(A)

dy/(dx )=(2e^x)/(e^x+2)

At x=0, dy/dx=(2e^0)/(e^0+2)=2/3

Equation of tangent line at x=0: y-2ln3=2/3 (x-0)

y-2ln3=2/3 x or y=2/3 x+2ln3

(B)

f(0.3)≈2/3 (0.3)+2ln3≈2.39722≈2.397

(C)

dy/dx=〖2e〗^x/(e^x+2) → dy=(2e^x)/(e^x+2) → ∫dy= ∫〖(2e^x)/(e^x+2) dx〗

Let u=e^x+2, du=e^x dx
2du=2e^x

Here is where I run into problems. Can anybody help me with the rest? Any help would be greatly appreciated.
 
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  • #2
bump. please, I'm desperate for help.
 
  • #3
helpmeimdumb said:

Homework Statement



1. The slope of a function at any point (x,y) is 〖2e〗^x/(e^x+2) . The point (0,2ln3) is on the graph of f.

(A) Write an equation of the tangent line to the graph of f at x=0.
(B) Use the tangent line in part A to approximate f(0.3) to the nearest thousandth.
(C) Solve the differential equation dy/(dx )=(2e^x)/(e^x+2) with the initial condition f(0)=2ln3.
(D) Use the solution in part C to find f(0.3) to the nearest thousandth.

Homework Equations



The Attempt at a Solution



(A)

dy/(dx )=(2e^x)/(e^x+2)

At x=0, dy/dx=(2e^0)/(e^0+2)=2/3

Equation of tangent line at x=0: y-2ln3=2/3 (x-0)

y-2ln3=2/3 x or y=2/3 x+2ln3

(B)

f(0.3)≈2/3 (0.3)+2ln3≈2.39722≈2.397

(C)

dy/dx=〖2e〗^x/(e^x+2) → dy=(2e^x)/(e^x+2) → ∫dy= ∫〖(2e^x)/(e^x+2) dx〗

Let u=e^x+2, du=e^x dx
2du=2e^x

Here is where I run into problems. Can anybody help me with the rest? Any help would be greatly appreciated.

Rewrite your integral in terms of the variable, u .
 

1. What is a differential equation?

A differential equation is an equation that involves a function and its derivatives. It relates the rates of change of the function to the function itself. This type of equation is commonly used to describe physical phenomena in fields such as physics, engineering, and biology.

2. Why is solving a differential equation important?

Solving a differential equation allows us to understand and predict the behavior of a system over time. This is crucial in fields like physics and engineering where accurate predictions are necessary for designing and implementing systems.

3. What are the different methods for solving a differential equation?

There are several methods for solving a differential equation, including separation of variables, substitution, and using integrating factors. The appropriate method depends on the type of differential equation and its initial conditions.

4. What is the role of initial conditions in solving a differential equation?

Initial conditions are the values of the function and its derivatives at a specific point. These conditions are necessary for finding a unique solution to a differential equation. Without them, the equation would have an infinite number of solutions.

5. Can differential equations be solved analytically?

Some differential equations can be solved analytically, meaning an exact solution can be found using mathematical operations. However, many differential equations are too complex to be solved analytically and require numerical methods to find approximate solutions.

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