Question

Form the differential equations from the following primitives where constants are arbitrary:
(i) y² = 4ax
(ii) y = cx + 2c² + c³
(iii) xy = a²
(iv) y = ax² + bx + c

Answer 1

(i) The equation of family of curves is
$y^2=4ax$ ...(1)
where $a$ is an arbitrary constant.
This equation contains only one arbitrary constant, so we shall get a differential equation of first order.
Differentiating equation (1) with respect to $x$, we get
$$\begin{gathered} 2y\frac{dy}{dx}=4a \\ \Rightarrow \frac{y}{2}\frac{dy}{dx}=a\left(2\right) \end{gathered}$$
Putting the value of $a$ in equation (1), we get
$$\begin{gathered} y^2=4\frac{y}{2}\frac{dy}{dx}x \\ \Rightarrow y=2x\frac{dy}{dx}, \\ \mathrm{It}\mathrm{is}\text{the required differential equation.} \end{gathered}$$

(ii) The equation of family of curves is
$y=cx+2c^2+c^3$ ...(1)
where $c$ is an arbitrary constant.
This equation contains only one arbitrary constant, so we shall get a differential equation of first order.
Differentiating equation (1) with respect to $x$, we get
$\frac{dy}{dx}=c$ ...(2)
Putting the value of $c$ in equation (1), we get
$y=x\frac{dy}{dx}+2{\left(\frac{dy}{dx}\right)}^2+{\left(\frac{dy}{dx}\right)}^3$
It is the required differential equation.

(iii) The equation of family of curves is
$xy=a^2$ ...(1)
where $a$ is an arbitrary constant.
This equation contains only one arbitrary constant, so we shall get a differential equation of first order.
Differentiating equation (1) with respect to $x$, we get
$y+x\frac{dy}{dx}=0$
It is the required differential equation.

(iv) The equation of family of curves is
$y=a{x}^2+bx+c$ ...(1)
where $a,b\mathrm{and}c$ are arbitrary constants. So, we shall get a differential equation of third order.
Differentiating equation (1) with respect to $x$, we get
$\frac{dy}{dx}=2ax+b$ ...(2)
Differentiating equation (2) with respect to $x$, we get
$\frac{d^{2}y}{d{x}^2}=2a$ ...(3)
Differentiating equation (3) with respect to $x$, we get
$\frac{d^{3}y}{d{x}^3}=0$
It is the required differential equation.