Solutions for Linear Equations (Section 1.5) Selected Problems

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.

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