calc 2 questions - integration and differential equation

[killasnake]

Hi, I am taking a calc 2 class right now, and I am just confused with this integration and differetial equation problem. I have no idea what to do.

Perform the following integration:

and

Solve the differential equation with given initial condition:

\(\displaystyle y=\)_________

 

[galactus]

Hi, I am taking a calc 2 class right now, and I am just confused with this integration and differetial equation problem. I have no idea what to do.

Perform the following integration:

Your integral is a hyperbolic identity. It is equivalent to:

\(\displaystyle \L\\\int_{0}^{tanh(2)}(tanh(2x))dx\)

[/quote]


and

Solve the differential equation with given initial condition:

\(\displaystyle y=\)_________

Separate variables:

\(\displaystyle \L\\\frac{dy}{dt}=\frac{y}{2}\)

\(\displaystyle \L\\\frac{dy}{y}=\frac{1}{2}dt\)

Integrate:

\(\displaystyle \L\\\int\frac{dy}{y}=\int\frac{1}{2}dt\)

\(\displaystyle \L\\ln(y)=\frac{t}{2}+C\)

\(\displaystyle \L\\y=C_{1}e^{\frac{t}{2}}\)

Now, use the initial condition to finish.