Find the solution of the initial value problem. $$ \frac{d P}{d t}=20 e^{t}, \quad P(0)=10 $$ $$ P= $$

Question

UpStudy AI · Accepted Answer

We start with the differential equation

$$
\frac{dP}{dt} = 20e^t
$$

and the initial condition

$$
P(0) = 10.
$$

**Step 1: Integrate the differential equation.**

Integrate both sides with respect to $$ t $$:

$$
\int \frac{dP}{dt} \, dt = \int 20e^t \, dt.
$$

This yields

$$
P(t) = 20 \int e^t \, dt = 20e^t + C,
$$

where $$ C $$ is the constant of integration.

**Step 2: Apply the initial condition to find $$ C $$.**

Substitute $$ t = 0 $$ and $$ P(0) = 10 $$ into the solution:

$$
10 = 20e^0 + C.
$$

Since $$ e^0 = 1 $$, we have

$$
10 = 20 + C.
$$

Thus,

$$
C = 10 - 20 = -10.
$$

**Step 3: Write the final solution.**

Substitute $$ C = -10 $$ into the expression for $$ P(t) $$:

$$
P(t) = 20e^t - 10.
$$
$$ P(t) = 20e^{t} - 10 $$

Find the solution of the initial value problem. \( \frac{d P}{d t}=20 e^{t}, \quad P(0)=10 \) \( P= \)