Question

The particular integral of \(\frac{d^2y}{dx^2} + 3\frac{dy}{dx} + 2y = e^{-2x}\) is

• A. \(-xe^{-2x}\)
• B. \(xe^{-2x}\)
• C. \(-\frac{x}{2}e^{-2x}\)
• D. \(\frac{x}{2}e^{-2x}\)

Hint:

When finding the particular integral for \(e^{ax}\) using the operator method, always first check if 'a' is a root of the auxiliary equation. If it is (i.e., if \(f(a)=0\)), you know it's a case of failure and you must apply the rule of multiplying by x and differentiating the denominator.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question asks about the ion-exchange mechanism in the permutit process for water softening, specifically which ions are removed from hard water.

Step 2: Key Formula or Approach (Alternate):
Think of permutit as a sodium-rich zeolite (Na₂Z) that swaps its Na⁺ ions for the hardness-causing divalent cations. The zeolite framework has a stronger attraction for divalent ions over monovalent ones.

Step 3: Detailed Explanation:
Hard water contains Ca²⁺ and Mg²⁺ ions from dissolved salts like chlorides and sulfates. When passed through permutit (sodium aluminum silicate), the Ca²⁺ and Mg²⁺ displace Na⁺ from the zeolite matrix due to higher charge density and stronger electrostatic attraction. The reactions are: Ca²⁺ + Na₂Z → CaZ + 2Na⁺ and Mg²⁺ + Na₂Z → MgZ + 2Na⁺. The outgoing water now contains harmless Na⁺ instead of hardness ions. Once the zeolite is exhausted (all Na⁺ replaced), it is regenerated by flushing with concentrated brine (NaCl solution), reversing the exchange.

Step 4: Final Answer:
Na⁺ ions from permutit are exchanged with Ca²⁺ and Mg²⁺ ions present in hard water.