Resolution of first order linear differential equations

Let's solve the first order linear differential equation dy/dt + p(t)y = g(t). Solving such an equation is to find y. The first thing we have to do is to find a magical function μ(t) called the integrating factor. We multiply both sides of the equation by this function. A subsequent set of steps will hopefully allow to find y. Rather than solving the general form let's solve a numerical example and then we will generalize the process to the solution of any first linear differential equation.

Let's solve the equation dv/dt = 9.8 - 0.196v
Let's put this equation in its general form: dv/dt + 0.19v = 9.8.
Let's find the magical function  μ(t). The solution of the general form of the first linear differential equation above defines this function as:

In the differential equation p(t) = 0.19 therefore

Let's multiply both sides of the equation by the value of μ(t)

The left side of the equation is the derivative of the product

Therefore we have

Let's integrate both sides of the equation we have:

We generalize the procedure used to solve the differential equation above. In order to solve a first linear differential equation having the form dy/dt + p(t)y = g(t) we follow the following steps:
1) We make sure that the differential equation is written in its proper explicit form
2) We find the integrating factor

3) We multiply both sides of the equation by μ(t)
4) We substitute the left side of the equation by the derivative of a product
5) We integrate both sides of the equation in order to find the unknown

Practice. Solve the following differential equation dy/dx = 3 + 2.5y

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