AP Calculus AB Exam Question

Consider the differential equation:

$\frac{dy}{dx} = 2x$

1. Using separation of variables, find the general solution of the given differential equation.

2. Find the particular solution to the differential equation with the initial condition $y(0) = 1$.

Answer

1. To solve the given differential equation using separation of variables, we first write it in the form:

$\frac{dy}{dx} = 2x$

Now, we can separate the variables by multiplying both sides of the equation by $dx$ and dividing both sides by $x$:

$\frac{dy}{x} = 2x \ dx$

Next, we can integrate both sides of the equation separately. The integral on the left side is with respect to $y$, and the integral on the right side is with respect to $x$:

$\int \frac{dy}{x} = \int 2x \ dx$

To evaluate the integral on the left side, we use the natural logarithm function:

$ln|y| = x^2 + C_1$

where $C_1$ is the constant of integration.

2. Now, to find the particular solution with the initial condition $y(0) = 1$, we substitute $x = 0$ and $y = 1$ into the general solution:

$ln|1| = 0^2 + C_1$

$0 = C_1$

Hence, the particular solution to the differential equation with the initial condition $y(0) = 1$ is:

$ln|y| = x^2$

$y = e^{x^2}$

Therefore, the particular solution to the given differential equation with the initial condition $y(0) = 1$ is $y = e^{x^2}$.