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- 1. <ul><li>These equations can easily be solved after </li></ul><ul><li>separate the variables </li></ul>DE with separable variables.
- 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. 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
- 10. gives the solution:
- 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
- 13. , gives So the particular solution is: We now use the information which means and
- 14. Exercise Solve

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