Final answer:
The general solution of the differential equation
is
, where
is the constant of integration. The corresponding particular solution, given the initial condition
, is
.
Step-by-step explanation:
To find the general solution of the given differential equation
, we can use the method of separation of variables. Rearranging the equation, we have:

Multiplying both sides of the equation by
, we get:

This is a linear first-order differential equation. We can separate the variables by moving all terms involving
to one side and all terms involving
to the other side:

Next, we can divide both sides of the equation by
to isolate
:

Now, we can integrate both sides of the equation with respect to


Simplifying the integrals, we have:

Integrating the right side of the equation, we get:

Using the power rule of integration, we can evaluate the integral:

where
is the constant of integration.
This is the general solution of the differential equation
.
To find the corresponding particular solution, we can use the given initial condition
. Substituting
and
into the general solution, we have:

Simplifying the equation, we get:

Solving for
, we find:

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

This is the general solution and the corresponding particular solution of the given differential equation.