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: Defining Linearity:
A first-order differential equation is categorized as linear if it can be rearranged into the standard formats \(\frac{dy}{dx} + P(x)y = Q(x)\) or \(\frac{dx}{dy} + P(y)x = Q(y)\). In these standard forms, the dependent variable (y or x, respectively) and its first derivative appear only to the power of one and are not multiplied together.
Step 3: Equation Analysis:
We examine each equation:


(A) $\frac{dy{dx} + P(x)y = Q(x)$:} This equation precisely matches the definition of a linear first-order differential equation with y as the dependent variable. Thus, (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 yields \(\frac{dy}{dx} = \frac{x+2y}{x-y}\). This equation cannot be manipulated into either of the standard linear forms. It involves products of y and \(\frac{dy}{dx}\), and it is a homogeneous equation, not a linear one.

(D) $(1 + x^2)\frac{dy{dx} + 2xy = 2$:} To ascertain linearity, we attempt to convert it to the standard form. Dividing the entire equation by \((1 + x^2)\) gives:
\[ \frac{dy}{dx} + \frac{2x}{1 + x^2}y = \frac{2}{1 + x^2} \] This equation conforms directly to the structure \(\frac{dy}{dx} + P(x)y = Q(x)\), with \(P(x) = \frac{2x}{1 + x^2}\) and \(Q(x) = \frac{2}{1 + x^2}\). Therefore, (D) is classified as a linear differential equation.

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