Help With Solving This Differential Equation
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Feb27-12, 11:28 PM
1. The problem statement, all variables and given/known data
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.
2. Relevant equations
3. The attempt at a solution
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
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
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 .