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# Lecture 2

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### Lecture 2

1. 1. <ul><li>These equations can easily be solved after </li></ul><ul><li>separate the variables </li></ul>DE with separable variables.
2. 2. <ul><li>These equations cannot be solved by separating </li></ul><ul><li>the variables, because the variables are un </li></ul><ul><li>-separable. These are called linear first-order DE. </li></ul>Non-separable variables.
3. 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. 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. 5. The resulting equation, Is then easy to solve, not because it’s exact, but because the left-hand side collapse.
6. 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.
7. 8. Exercise 1 Solve Solution: This is already in the required form
8. 9. With and The integrating factor is Thus the integrating factor is . Multiplying both sides of the equation by
9. 10. gives the solution:
10. 11. <ul><li>Solve and that when . </li></ul>First we change the equation to the required form: with and . The integrating factor is Example 2 Solution
11. 13. , gives So the particular solution is: We now use the information which means and
12. 14. Exercise Solve