Answer
\( y(x) = F(x) + y_0 - F(x_0) \), where \( F(x) \) is any antiderivative of \( f(x) \).
Work Step by Step
The Fundamental Theorem of Calculus provides a solution to the initial value problem \( \frac{dy}{dx} = f(x) \), \( y(x_0) = y_0 \), where \( f \) is continuous, by connecting the indefinite integral of \( f(x) \) to the solution of the differential equation.
Specifically, let \( F(x) \) be any antiderivative of \( f(x) \), then by the Fundamental Theorem of Calculus, the definite integral of \( f(x) \) from \( x_0 \) to \( x \) is \( F(x) - F(x_0) \).
Now, define \( y(x) = F(x) + C \), where \( C \) is an arbitrary constant. By differentiating \( y(x) \), we get \( \frac{dy}{dx} = f(x) \), which satisfies the given differential equation.
To determine the value of \( C \), we use the initial condition \( y(x_0) = y_0 \), which gives us \( F(x_0) + C = y_0 \), hence \( C = y_0 - F(x_0) \).
Therefore, the solution to the initial value problem is \( y(x) = F(x) + y_0 - F(x_0) \), where \( F(x) \) is any antiderivative of \( f(x) \).
