# Solutions for linear equation s (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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