Show that y=4ax is a solution of the differential equation y=x dy… - Vidyadip

Question

Show that $\text{y}=4\text{ax}$ is a solution of the differential equation $\text{y}=\text{x}\frac{\text{d}\text{y}}{\text{dx}}+\text{a}\frac{\text{dx}}{\text{dy}}.$

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Answer

We have,

$\text{y}=4\text{ax}\ ...(1)$

Differentiating both sides of equation (1) with respect to 3, we get

$2\text{y}\frac{\text{dy}}{\text{dx}}=4\text{a}$

$\Rightarrow\frac{\text{dy}}{\text{dx}}=\frac{2\text{a}}{\text{y}}\ ...(2)$

now, differentiating both sided of (1) with respect to y, we get

$2\text{y}=4\text{a}\frac{\text{dy}}{\text{dx}}$

$\Rightarrow\frac{\text{dy}}{\text{dx}}=\frac{\text{y}}{2\text{a}}\ ...(3)$

$\therefore\text{x}\frac{\text{dy}}{\text{dx}}+\text{a}\frac{\text{dy}}{\text{dx}}=\text{x}\Big(\frac{2\text{a}}{\text{y}}\Big)+\text{a}\Big(\frac{\text{y}}{2\text{a}}\Big)$

$\Rightarrow\text{x}\frac{\text{dy}}{\text{dx}}+\text{a}\frac{\text{dy}}{\text{dx}}=\frac{2\text{ax}}{\text{y}}+\frac{\text{y}}{2}$

$\Rightarrow\text{x}\frac{\text{dy}}{\text{dx}}+\text{a}\frac{\text{dy}}{\text{dx}}=\frac{\text{y}^2}{2\text{y}}+\frac{\text{y}}{2}$

$\Rightarrow\text{x}\frac{\text{dy}}{\text{dx}}+\text{a}\frac{\text{dy}}{\text{dx}}=\frac{\text{y}}{2}+\frac{\text{y}}{2}$

$\Rightarrow\text{x}\frac{\text{dy}}{\text{dx}}+\text{a}\frac{\text{dy}}{\text{dx}}=\text{y}$

$\Rightarrow\text{y}=\text{x}\frac{\text{dy}}{\text{dx}}+\text{a}\frac{\text{dy}}{\text{dx}}$

Hence, the given function is the solution to the given differential equation.

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