| Problem - You want to integrate | |
| (1) |
| However, that f(z)y term really messes things up! If you only had an expression of the form | |
| (2) |
things would be much easier, then you could integrate with respect to z and find y(z). How can you combine f(z) and y to get this simplification? First note that  is of the form of the derivative of a product, so examine first the product y u, where u is some function of f(z) you still have to define. Recall
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| (3) |
| That's looking close to the left hand side of equation (1), but there is a "u" in front of the dy/dz term, and a du/dz expression where f(z) is. If you had a form of u such that du/dz = u f(z), then you could manipulate equation (3): | |
| (4) |
| where the term in brackets is the left hand side of equation (1). You need du/dz = u f(z). Recall | |
| (5) |
If you define  and f(z) = dq/dz (i.e.  , then
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| (6) |
This satisfies the condition that du/dz=u f(z). That's the ezpression you needed! Therefore,  , and substituting into equation (4),
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| (7) |
| where the term in brackets is the left hand side of equation (1). CONCLUSION: If your problem is of the form | |
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| (1) |
| you can multiply both sides of the equation by the | |
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| (which you should be able to evaluate, since you know f(z)), to yield | |
| (9) |
| or, substituting from equation (7) | |
| (10) |
| so that | |
| (11) |
| (12) |
| EXAMPLE | |
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| f(t)=k2, so | |
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| From equation (12), the solution is then | |
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| The constant can be obtained from the intitial condition that at t=0, CB=0; | |
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