First Order Linear Equations
A first order linear differential equation has the following form:
The general solution is given by
where
called the integrating factor. If an initial condition is given, use it to find the constant C.
Here are some practical steps to follow:
1.
If the differential equation is given as
,
rewrite it in the form
,
where
2.
Find the integrating factor
.
3.
Evaluate the integral
4.
Write down the general solution
.
5.
If you are given an IVP, use the initial condition to find the constant C.
Example: Find the particular solution of:
Solution: Let us use the steps:
Step 1: There is no need for rewriting the differential equation. We have
Step 2: Integrating factor
.
Step 3: We have
.
Step 4: The general solution is given by
.
Step 5: In order to find the particular solution to the given IVP, we use the initial condition to find C.
Indeed, we have
.
Therefore the solution is
.
Note that you may not have to do the last step if you are asked to find the general solution (not an IVP).
Linear Equations: Answer to Example 1
Example: Find the solution to
.
Solution: First, we recognize that this is a linear equation. Indeed, we have
Therefore, the integrating factor is given by
.
Since
, we get
Hence, the general solution is given by the formula
We have
The details for this calculation involve the technique of integrating rational functions. We have
Hence, the only difficulty is in the integral
differentiate t and integrate
. Here we will use integration by parts. We will
. The details are left to the reader. We have
Therefore, we have
,
which clearly implies
The general solution can also be rewritten as
Finally, the initial condition y(0) = 0.4 gives C = 0.4. Therefore, the solution to the IVP is
Linear Equations: Answer to Example 2
Example: Solve the following initial value problem for t > 0
Answer: This is a linear equation. Let us follow these steps for solving such equations:
1.
We have to divide by 2t
2.
We get the integration factor u(t) by
3.
The general solution is given by
.
Since
Therefore, we have
4.
The solution to the given initial value problem may be obtained by using the initial condition y(2)=4. We
have
,
which gives
. Therefore, the solution is
Linear Equations: Answer to Example 3
Example: Find the solution to
Answer: This is a linear equation. First we have to rewrite the equation with no function in front of y'. We get
,
which may also be rewritten as
.
Hence, the integrating factor is given by
Therefore, the general solution can be obtained as
Since we have
we get
The initial condition
implies
,
which gives C=-1. Therefore, the particular solution to the initial value problem is
Source: sosmath.com