Differential Equations - dy/dx = g(y)

An equation that involves independent variables, dependent variables, derivatives of the dependent variables with respect to independent variables, and constant is called a differential equation.

The case of \(\frac{dy}{dx}=g(y)\) is very similar to the method of \( \frac{dy}{dx}=f(x).\) We will look at some examples in a moment, but in this case we may need a little bit more rearranging and algebra involved.

Let's look at some examples:

If \(\frac{dy}{dx}=y^2,\) express \(y\) in terms of \(x.\)

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This differential equation is separable—we can move the \(dy\) and \(dx\) around and then integrate both sides to find a general solution.

Multiply both sides by \({dx}\) and divide by \(y^{2},\) which gives us \[\frac{dy}{y^2}={dx}.\] Now, taking the integral of both sides, we have \[\int\frac{dy}{y^2}=\int{dx},\] which now becomes the following if we integrate: \[-\frac{1}{y} + c_1=x + c_2 \quad (c_1,c_2 \in \mathbb{R}), \] and by rearranging we get \[y=-\frac{1}{x+c} \quad (c \equiv c_2-c_1).\ _\square\]

If \(\frac{dy}{dx}=y^2+2,\) express \(y\) in terms of \(x.\)

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Again this is separable, so we can do the same as in the previous example, by multiplying both sides by \({dx}\) and dividing by \({y^2+2}:\) \[\frac{dy}{y^2+2}={dx}.\] Taking the integral of both sides gives \[\int\frac{dy}{y^2+2}=\int{dx},\] which now becomes the following if we integrate: \[\dfrac{\sqrt{2}}{2}\arctan{ \left ( \dfrac{\sqrt{2}}{2}y \right ) }=x+c,\] and we can now rearrange to get \[y=\sqrt2\tan\big(\sqrt2(x+c)\big).\ _\square\]