Question

Which of the following are linear first order differential equations?
(A) $\frac{dy}{dx} + P(x)y = Q(x)$
(B) $\frac{dx}{dy} + P(y)x = Q(y)$
(C) $(x - y)\frac{dy}{dx} = x + 2y$
(D) $(1 + x^2)\frac{dy}{dx} + 2xy = 2$
Choose the correct answer from the options given below:

• A. (A), (B) and (D) only
• B. (A) and (B) only
• C. (A), (B) and (C) only
• D. (A), (B), (C) and (D)

Hint:

To test for linearity, always try to rearrange the equation into one of the two standard forms. If the dependent variable or its derivative appears with a power other than one, or in a non-linear function (like sin(y)), or are multiplied together, the equation is non-linear.

Answer 1

The Correct Option is A

Step 1: Concept Definition:
A first-order differential equation is classified as linear if it can be expressed in either the standard form \(\frac{dy}{dx} + P(x)y = Q(x)\) or \(\frac{dx}{dy} + P(y)x = Q(y)\). Within these forms, the dependent variable (which is y in the first case and x in the second) and its derivative must appear only to the first power and must not be multiplied together.
Step 3: Detailed Analysis:
Let's examine each equation:


(A) $\frac{dy{dx} + P(x)y = Q(x)$:} This equation directly matches the definition of a linear first-order differential equation with y as the dependent variable. Therefore, (A) is linear.

(B) $\frac{dx{dy} + P(y)x = Q(y)$:} This equation is the standard representation of a linear first-order differential equation with x as the dependent variable. Consequently, (B) is linear.

(C) $(x - y)\frac{dy{dx} = x + 2y$:} Rearranging this equation yields \(\frac{dy}{dx} = \frac{x+2y}{x-y}\). This equation cannot be manipulated into either of the standard linear forms. It includes terms where y and \(\frac{dy}{dx}\) are multiplied, and it is a homogeneous equation, not a linear one.

(D) $(1 + x^2)\frac{dy{dx} + 2xy = 2$:} To determine if this is linear, we will attempt to convert it to the standard form. Divide the entire equation by \((1 + x^2)\):
\[ \frac{dy}{dx} + \frac{2x}{1 + x^2}y = \frac{2}{1 + x^2} \] This equation conforms precisely to the structure \(\frac{dy}{dx} + P(x)y = Q(x)\), with \(P(x) = \frac{2x}{1 + x^2}\) and \(Q(x) = \frac{2}{1 + x^2}\). Thus, (D) is a linear differential equation.

Step 4: Conclusion:
The equations identified as linear first-order differential equations are (A), (B), and (D). This corresponds to option (1).