Separable differential equations - A few illustrating examples

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Example 1

Say we have

$\frac{{dy}}{{dx}} = \frac{y}{x}$

So the slopes should be equal to the positions? That ought to be lines through the origin. Let us see.

We get

$\frac{{dy}}{y} = \frac{{dx}}{x}$

Integrating this gives us

$\ln |y| = \ln |x| + {C_1}$

$|y| = {e^{\ln |x| + {C_1}}} = {e^{\ln |x|}}{e^{{C_1}}} = {C_2}|x|$

Yet again, as in the previous page, the signs from the absolute values can be absorbed into the constant to finally give us

$y = Cx$

I.e. straight lines through the origin.

Example 2

$\frac{{dy}}{{dx}} = - \frac{x}{y}$

Ok, so the slope is now the negative reciprocal of the position. What could that give us? Let us see. We get

$ydy = - xdx$

or, after integrating,

$\frac{{{y^2}}}{2} = - \frac{{{x^2}}}{2} + {C_1}$

Multiplying through by 2 and rearranging a bit gives us

$\frac{{{y^2}}}{2} = - \frac{{{x^2}}}{2} + {C_1}$

In this we should recognise the equation of a circle. Changing the constant to emphasize that we get

${x^2} + {y^2} = {r^2}$

Below we can see a graph of the slope field and a few selected solutions.

It should hopefully make sense that the derivative is the negative reciprocal of the position for these circles that are concentric around the origin. The tangent property of circles will give us that a radial line through a point on a circle, and the tangent line through the same point will have slopes that are the negative reciprocals of each other.

Example 3

$\frac{{dy}}{{dx}} = - \frac{y}{x}$

We get

$\frac{{dy}}{y} = - \frac{{dx}}{x}$

$\int {\frac{{dy}}{y}} = - \int {\frac{{dx}}{x}}$

and thus

$\ln |y| = - \ln |x| + {C_1}$

$\ln |y| = \ln \left| {\frac{1}{x}} \right| + {C_1}$

This gives us

$|y| = {e^{\ln \left| {\frac{1}{x}} \right| + {C_1}}} = \left| {\frac{1}{x}} \right|{e^{{C_1}}} = \frac{{{C_2}}}{{\left| x \right|}}$

Yet again we let the constant absorb the signs. This finally gives us

$y = \frac{C}{x}$

A slope field with a few particular solutions is shown below.

By the way, as you have seen I usually rename the constant using subscripts as it changes throughout the calculations. The final one is usually written without a subscript.

Example 4

If we follow the patterns of the examples we have left to examine

$\frac{{dy}}{{dx}} = \frac{x}{y}$