Differential Equations

The order of a differential equation

The order of a differential equation is simply the order of the highest derivative appearing in it. No need to solve anything first.

What is the order of $2x^2\dfrac{d^2y}{dx^2} - 3\dfrac{dy}{dx} + y = 0$ ?

Solution: The highest derivative present is $\dfrac{d^2y}{dx^2}$ , a second derivative, so the order is 2.

Form the differential equation representing the family of curves $y = mx$ , where $m$ is an arbitrary constant.

Solution: Differentiate $y = mx$ with respect to $x$ :

\dfrac{dy}{dx} = m

Since $m = \dfrac{y}{x}$ from the original equation, substituting eliminates the arbitrary constant, giving the differential equation:

\dfrac{dy}{dx} = \dfrac{y}{x}

Find the equation of the curve passing through $(-2, 3)$ , given that the slope of the tangent at any point $(x, y)$ is $\dfrac{2x}{y^2}$ .

Solution: The slope is $\dfrac{dy}{dx} = \dfrac{2x}{y^2}$ . Separating variables and integrating both sides:

y^2\,dy = 2x\,dx \;\Rightarrow\; \int y^2\,dy = \int 2x\,dx \;\Rightarrow\; \dfrac{y^3}{3} = x^2 + C

Using the point $(-2, 3)$ to find $C$ :

\dfrac{27}{3} = 4 + C \;\Rightarrow\; 9 = 4 + C \;\Rightarrow\; C = 5

So $\dfrac{y^3}{3} = x^2 + 5$ , or equivalently:

y^3 = 3x^2 + 15