Questions: Question 3 Solve the initial value problem dy/dx=e^(-x), y(0)=0. e^(-x)-1 e^(-x) e^(-x)+1 -e^(-x)-1 -e^(-x) -e^(-x)+1

Transcript text: Question 3 Solve the initial value problem $\frac{d y}{d x}=e^{-x}, y(0)=0$. $e^{-x}-1$ $e^{-x}$ $e^{-x}+1$ $-e^{-x}-1$ $-e^{-x}$ $-e^{-x}+1$

Solution

Solution Steps

To solve the initial value problem $\frac{d y}{d x}=e^{-x}$ with $y(0)=0$, we need to integrate the differential equation and then apply the initial condition to find the constant of integration.

Step 1: Solve the Differential Equation

We start with the differential equation given by

\[ \frac{dy}{dx} = e^{-x}. \]

To find the general solution, we integrate both sides with respect to $x$:

\[ y = \int e^{-x} \, dx = -e^{-x} + C, \]

where $C$ is the constant of integration.

Step 2: Apply the Initial Condition

Next, we apply the initial condition $y(0) = 0$:

\[ 0 = -e^{0} + C \implies 0 = -1 + C \implies C = 1. \]

Step 3: Write the Particular Solution

Substituting $C$ back into the general solution, we obtain the particular solution:

\[ y = -e^{-x} + 1. \]