Lecture 2
- 1. These equations can easily be solved after separate the variables DE with separable variables.
- 2. These equations cannot be solved by separating the variables, because the variables are un -separable. These are called linear first-order DE. Non-separable variables.
- 3. Linear First-Order Differential Equations A first-order differential equation is said to be linear if it can be expressed in the form : Where and are functions of x.
- 4. To solve a first-order linear equation, first rewrite it (if necessary) in the standard form above then multiply both sides by the integrating factor
- 5. The resulting equation, Is then easy to solve, not because it’s exact, but because the left-hand side collapse.
- 7. Therefore, the general equation becomes Making it susceptible to an integration, which gives the solution : Do not memorize this equation for the solution ; memorize the step needed to get there.
- 8. Exercise 1 Solve Solution: This is already in the required form
- 9. With and The integrating factor is Thus the integrating factor is . Multiplying both sides of the equation by
- 11. Solve and that when . First we change the equation to the required form: with and . The integrating factor is Example 2 Solution
- 13. , gives So the particular solution is: We now use the information which means and
- 14. Exercise Solve