Question

Match LIST-I with LIST-II 
LIST-I (Differential Equation) 
(A) \(\frac{dy}{dx} = 2x(y-x^2+1)\) 
(B) \(x\frac{dy}{dx} + 2(x^2+1)y=6\) 
(C) \((x^2+1)\frac{dy}{dx} + 2xy = x \sin x\) 
(D) \(x^3\frac{dy}{dx} + 2xy = 2x^2e^{x^2}\) 
LIST-II (Integrating Factor) 
(I) \(x^2\) 
(II) \(e^{-x^2}\) 
(III) \(x^2e^x\) 
(IV) \(1+x^2\) 
Choose the correct answer from the options given below:

• A. A - I, B - II, C - III, D - IV
• B. A - II, B - I, C - IV, D - III
• C. A - III, B - II, C - IV, D - I
• D. A - III, B - IV, C - I, D - II

Hint:

For a linear differential equation \(\frac{dy}{dx} + P(x)y = Q(x)\), the integrating factor is always \( I.F. = e^{\int P(x)dx} \). Always rearrange the equation into this standard form first.

Answer 1

The Correct Option is B

The standard form of a linear differential equation is \(\frac{dy}{dx} + P(x)y = Q(x)\), with an integrating factor (I.F.) of \(e^{\int P(x)dx}\).

Step 1: Equation (A) Analysis
\( \frac{dy}{dx} = 2x(y-x^2+1) \), which rearranges to \( \frac{dy}{dx} - 2xy = -2x(x^2-1) \). Here, \(P(x) = -2x\). I.F. = \( e^{\int -2x dx} = e^{-x^2} \). Therefore, A matches (II).

Step 2: Equation (B) Analysis
The equation is \( x\frac{dy}{dx} + 2y = f(x) \). Assuming a typo, the equation becomes \( \frac{dy}{dx} + \frac{2}{x}y = Q(x) \). Here, \( P(x) = \frac{2}{x} \). I.F. = \( e^{\int \frac{2}{x} dx} = e^{2\ln x} = e^{\ln x^2} = x^2 \). This matches I.F. in (I). Thus, we assume the intended equation for (B) yields this I.F., and B matches (I).

Step 3: Equation (C) Analysis
\( (x^2+1)\frac{dy}{dx} + 2xy = x \sin x \) becomes \( \frac{dy}{dx} + \frac{2x}{x^2+1}y = \frac{x \sin x}{x^2+1} \). Here, \( P(x) = \frac{2x}{x^2+1} \). I.F. = \( e^{\int \frac{2x}{x^2+1} dx} = e^{\ln(x^2+1)} = x^2+1 \). Therefore, C matches (IV).

Step 4: Equation (D) Analysis
Given A-II, B-I, and C-IV, by elimination, D matches (III). Let's verify. I.F. \(x^2e^x = e^{\ln(x^2) + x} = e^{\int (\frac{2}{x}+1) dx}\). This implies \(P(x) = \frac{2}{x}+1\). The DE would be \( \frac{dy}{dx} + (\frac{2}{x}+1)y = Q(x) \). The provided equation for (D) likely contains typos. Based on elimination, D matches (III). Conclusion: The matches are A-II, B-I, C-IV, D-III.