Question Statement

Solve the differential equation:

$\frac{dy}{dx}=\cos x$

subject to the initial condition $y(0)=1$.

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Background and Explanation

This problem requires solving a first-order ordinary differential equation using separation of variables. After finding the general solution, we apply the initial condition to determine the constant of integration and obtain the particular solution.

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Solution

We begin with the differential equation and initial condition:

$\frac{dy}{dx}=\cos x,y(0)=1$

Step 1: Separate the variables

Multiply both sides by $dx$ to separate the variables:

$dy=\cos xdx$

Step 2: Integrate both sides

Integrating both sides of the equation:

$\int dy=\int \cos xdx$

This yields the general solution:

$y=\sin x+c$

where $c$ is the constant of integration.

Step 3: Apply the initial condition

We are given that $y(0)=1$, which means when $x=0$, $y=1$. Substituting these values into equation (1):

$1=\sin (0)+c$

Since $\sin (0)=0$:

$1=0+c$

Therefore:

$c=1$

Step 4: Write the particular solution

Substituting $c=1$ back into equation (1), we obtain the final solution:

$y=\sin x+1$

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Key Formulas or Methods Used

• Separation of variables: Rearranging $\frac{dy}{dx}=f(x)$ into $dy=f(x)dx$ to isolate $y$ and $x$
• Integration of cosine: $\int \cos xdx=\sin x+C$
• Initial value problem method: Using the condition $y(x_0)=y_0$ to solve for the specific value of the integration constant

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Summary of Steps

1. Separate variables: $dy=\cos xdx$
2. Integrate both sides to obtain the general solution: $y=\sin x+c$
3. Substitute the initial condition $(x=0,y=1)$ to find the constant: $c=1$
4. Write the final particular solution: $y=\sin x+1$