Question 95 of 96beginner🧠 RememberMCQ1 mark

The general solution of a differential equation of the type dxdy+P1x=Q1\frac{dx}{dy} + P_1 x = Q_1 is

Exercise: Miscellaneous Exercise on Chapter 9 | Q: 14 | (Page 36)
For More Understanding

Explanation

According to the provided context, for a first order linear differential equation in the form dx/dy + P₁x = Q₁, where P₁ and Q₁ are constants or functions of y only, the Integrating Factor (I.F) = e^(∫P₁ dy). The general solution is given by: x · (I.F) = ∫(Q₁ × I.F) dy + C. Substituting the I.F, we get x · e^(∫P₁ dy) = ∫(Q₁ · e^(∫P₁ dy)) dy + C. Option C matches this formula exactly. Options A and B are incorrect because they have y on the left side instead of x. Option D is incorrect because it uses dx instead of dy in the integration.

Solution Steps

  1. Step 1: Identify that the differential equation is of the type dx/dy + P₁x = Q₁, where P₁ and Q₁ are functions of y only.

  2. Step 2: The Integrating Factor (I.F) = e^(∫P₁ dy).

  3. Step 3: The general solution is x · (I.F) = ∫(Q₁ × I.F) dy + C.

  4. Step 4: Substituting I.F: x · e^(∫P₁ dy) = ∫(Q₁ · e^(∫P₁ dy)) dy + C.

  5. Step 5: This matches Option (C).

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