Question

Find the DE whose solution is ${a^2}y - ax + 8 = 0$ .

Answer 1

Hint : The differential equation from its solution can be found by eliminating the arbitrary coefficient in it. It can be eliminated by differentiating the equation depending on the number of arbitrary constants present in it.

Complete step-by-step answer :
The given solution of the required differential equation is given as,
${a^2}y - ax + 8 = 0 \cdots \left( 1 \right)$
It is evident from the solution of the differential equation that it contains only one constant i.e.
To obtain the differential equation the value of the constant is to be determined.
The value of the constant can be obtained by differentiating the equation (1) concerning to.
Differentiating equation (1) concerning to x we get,
${a^2} \times \dfrac{{dy}}{{dx}} - a \times \dfrac{{d\left( x \right)}}{{dx}} + \dfrac{{d\left( 8 \right)}}{{dx}} = 0 \cdots \left( 2 \right)$
Equation (2) becomes as

$$\Rightarrow \dfrac{{dy}}{{dx}}{a^2} - a \times 1 + 0 = 0 \\\ \Rightarrow \dfrac{{dy}}{{dx}}{a^2} - a = 0 \cdots \left( 3 \right) \\\$$

$\dfrac{{dy}}{{dx}}$ can also be written as it signifies the first derivative. Substituting the value of $\dfrac{{dy}}{{dx}}$ in equation (3), we get
${a^2}{y_1} - a = 0 \cdots \left( 4 \right)$
Equation (4) is a quadratic equation in. Therefore, it has two solutions or two values of $x$ .
Solving equation (4), we get
$a\left( {a{y_1} - 1} \right) = 0$
Therefore, $a = 0$ and $a = \dfrac{1}{{{y_1}}}$ .
is not possible as the value of the arbitrary constant cannot be 0.
So, the acceptable solution is $a = \dfrac{1}{{{y_1}}}$ .
Substitute the value of $a = \dfrac{1}{{{y_1}}}$ in equation (1), we get

$$\Rightarrow {\left( {\dfrac{1}{{{y_1}}}} \right)^2}y - \left( {\dfrac{1}{{{y_1}}}} \right)x + 8 = 0 \\\ \Rightarrow \dfrac{y}{{{y_1}^2}} - \dfrac{x}{{{y_1}}} + 8 = 0 \cdots \left( 5 \right) \\\$$

Simplifying equation (5) by multiplying both sides by ${y_1}^2$ , we get
$y - x{y_1} + 8{y_1}^2 = 0$
Or it can be written by rearranging as
$8{y_1}^2 - x{y_1} + y = 0$
Thus, the required differential equation is $8{y_1}^2 - x{y_1} + y = 0$ whose solution is given as ${a^2}y - ax + 8 = 0$.

Note : The concept and definition of the differential should be clear. A differential equation is an equation consisting of differential coefficients of dependent variable concerning independent variable, along with dependent and independent variable.
The important thing in order to obtain the differential equation is to eliminate the arbitrary constants present in the original equation.
The equation should be differentiated as many times as there are constants present in the original equation. For instance if the number of constants is one it should be differentiated once, etc.