The solution of the differential equation \(\rm x \dfrac{dy}{dx}-y=3\) represents a family of:

Concept:

First Order Linear Differential Equation:
A differential equation of the from \(\rm \dfrac{dy}{dx}\) + P × y = Q, where P and Q are constants or functions of x only, is known as a first order linear differential equation.

Steps to solve a First Order Linear Differential Equation:

• Convert into the standard form \(\rm \dfrac{dy}{dx}\) + P × y = Q, where P and Q are constants or functions of x only.
• Find the Integrating Factor (F) by using the formula: F = \(\rm e^{\int P\ dx}\).
• Write the solution using the formula: \(\rm y\times F=\displaystyle \int (Q\times F)\ dx+C\) where C is the constant of integration.

Calculation:

\(\rm x \dfrac{dy}{dx}-y=3\)

\(\rm \Rightarrow \dfrac{dy}{dx}+\left(\dfrac{-1}{x}\right)y=\dfrac{3}{x}\)

⇒ P = \(\rm \dfrac{-1}{x}\) and Q = \(\rm \dfrac{3}{x}\).

Integrating factor F = \(\rm e^{\int P\ dx}=e^{\int \tfrac{-1}{x}\ dx}=e^{-\ln x}=\dfrac{1}{x}\).

The solution of the given differential equation is:

\(\rm y\times \dfrac{1}{x}=\displaystyle \int \left(\dfrac{3}{x}\times \dfrac{1}{x}\right)\ dx+C\)

\(\rm \Rightarrow \dfrac{y}{x}=3\left(\dfrac{-1}{x}\right)+C \)

⇒ y = Cx - 3, which is the equation of a straight line.