Separable, homogeneous, integrating factor and Euler, and how to recognise which equation is which.
Nearly all the difficulty here is choosing. Pick an equation and the method, the reason for it and the working appear together.
separable
| If it looks like | Use |
|---|---|
| dy/dx = (something in x) × (something in y) | separable |
| dy/dx = a function of y⁄x only | homogeneous, substitute y = vx |
| dy/dx + P(x)y = Q(x) | integrating factor |
| none of the above, or a numerical answer is wanted | Euler |
Test for homogeneous quickly: replace x with λx and y with λy. If every λ cancels, it is homogeneous and y = vx will work. That test takes ten seconds and saves you from attempting a separation that cannot happen.
For dydx + P(x)y = Q(x), multiply everything by e∫P dx. The left side then becomes the derivative of a product, by design, and you integrate both sides.
When nothing solves, step along the gradient: yn+1 = yn + h f(xn, yn), with xn+1 = xn + h.
| n | x | y | f = x + y | next y |
|---|---|---|---|---|
| 0 | 0 | 1 | 1 | 1.1 |
| 1 | 0.1 | 1.1 | 1.2 | 1.22 |
| 2 | 0.2 | 1.22 | 1.42 | 1.362 |
| 3 | 0.3 | 1.362 | 1.662 | 1.5282 |
| 4 | 0.4 | 1.5282 | 1.9282 | 1.72102 |
The exact solution here is y = 2ex − x − 1, giving 1.7974 at x = 0.5, so Euler is low by about 0.076. Keep every digit in the table and round only at the end.
1. Which method does dydx + 2yx = x³ need?
2. For dydx = xy with y(0) = 1, find y when x = 2, to 3 decimal places.
3. Using Euler with h = 0.1 on dy/dx = x + y, y(0) = 1, what is y₂?
Identifying the method and saying why. A wrong choice costs the whole question, and the right one is often worth a mark on its own.
On an integrating factor, showing e∫P dx simplified. On a homogeneous equation, stating y = vx and dy/dx = v + x dv/dx before substituting.
Using the initial condition at the end. A general solution with an unfound constant is incomplete whenever a condition was given.
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