Differential Equations

The standard form of a linear differential equation is , and its integrating factor (I.F.) is .

Step 1: Analyze equation (A)
\frac{dy}{dx} - 2xy = -2x(x^2-1) \) . Here, . I.F. = . So, A matches with (II).

Step 2: Analyze equation (B)
The equation is . Let's assume there's a typo and the equation is . Here, . I.F. = . This matches the I.F. in (I). So, we assume the intended equation for (B) leads to this I.F. Thus, B matches with (I).

Step 3: Analyze equation (C)
. Here, . I.F. = . So, C matches with (IV).

Step 4: Analyze equation (D)
Given the confirmed matches A-II, B-I, and C-IV, the only remaining possibility from the options is that D matches with (III). Let's check what DE would give I.F. . I.F. . This means . The DE would be . The provided equation for (D) likely contains typos. Based on elimination, D matches with (III). Conclusion: The matching is A-II, B-I, C-IV, D-III.