First Order Differential Equations

A first order differential equation involves a function y(x) and its first derivative dy/dx. In Further Mathematics, you need to solve several types: separable, linear (using an integrating factor), and use differential equations in modelling real-world problems.

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1 Separable Equations

Definition

A separable equation can be written in the form:

dy/dx = f(x) g(y)

This can be separated as:

(1/g(y)) dy = f(x) dx

Then integrate both sides.

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Worked Example 1

Solve dy/dx = xy, given y(0) = 2.

Solution:

Separate: (1/y) dy = x dx

Integrate: ln|y| = x^2/2 + C

So y = A e^(x^2/2) where A = e^C.

Apply y(0) = 2: 2 = A e^0 = A. So A = 2.

y = 2 e^(x^2/2)

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Worked Example 2

Solve dy/dx = (1 + y^2) / (1 + x^2).

Solution:

Separate: dy/(1 + y^2) = dx/(1 + x^2)

Integrate: arctan y = arctan x + C

So y = tan(arctan x + C).

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Worked Example 3

Solve x dy/dx = y(1 - y), given y(1) = 1/2.

Solution:

Separate: dy / (y(1-y)) = dx / x