For this problem p(x) = 3 and q(x) = 2x.
Step 1: 

Step 2: 
Step 3: 
Step 4: 


Step 5: 
This equation needs to be rewritten first. Dividing both
sides by x:
For this problem and
q(x) = sin(x).
Step 1: 

Step 2: 
Step 3: 
Step 4: 


Step 5: 
Step 1: 

Step 2: 
Step 3: 
Step 4: 
Step 5: 
First: Find all solutions (note that this is also a
separable equation).
Step 1: 

Step 2: 
Step 3: 
Step 4: 
Step 5: 
Second: Use the initial condition.
This is not a linear equation. It is a Riccati equation,
and if we knew a particular
solution, then we could make a change of variables to solve this equation.
This is not a linear equation. It is separable, however,
so we can solve this using
the technique for such equations.
Step 1: 

Step 2: 
Step 3: 

Step 4: 
In Step 1, we divided by
which is never zero .
There are no singular solutions. 
Step 1: 

Step 2: 
Step 3: 
Step 4: 


Step 5: 
First: Find all solutions.
Step 1: 

Step 2: 
Step 3: 
Step 4: 
Step 5: 
Second: Use the initial condition.
This is not a linear equation.
First: Find all solutions.
Step 1: 

Step 2: 
Step 3: 
Step 4: 
Step 5: 
Second: Use the initial condition.