We learn techniques to solve first-order linear and separable ODEs.

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### Examples

The ODE is separable.

- To separate, we move all factors involving to the left-hand side and all factors involving to the right-hand side. We also formally write as and verify that we can end up with on the left-hand side and on the right hand side:
- Apply an integral on both sides: That gives our general solution.
- If we had initial conditions , we would plug in , to our general solution to determine that . We can plug in this value for and solve for to determine that

The ODE is linear.

- To begin, we must write the ODE in standard form:
- We give the name to the coefficient of (note that it doesn’t
*have to*depend on . Writing is only meant to indicate that it*cannot*depend on ). We give the name to the right-hand side. - The integrating factor is the exponential of the integral of . In this case, we can take . Multiply both sides of the standard form equation by the integrating factor:
- The integrating factor gets its name because it is now possible to integrate the left-hand side. By the product rule, we know that which is exactly the right-hand side. Thus we have
- Integrating the left-hand side gives ; integrating both sides gives (don’t forget absolute values in the logarithm). Solving for gives as the general solution.