<p>We have \(y - (c_1 + c_2)\sin(x + c_3) - c_4 e^{x+c_5}\). The order of the differential equation whose general solution is given by this expression is:</p>
Step-by-Step Solution
Key Concept: The order of a differential equation equals the number of independent arbitrary constants in its general solution. Count the distinct independent constants: c₁, c₂, c₃, c₄, c₅ appear in the expression, but (c₁ + c₂) acts as a single parameter, and c₃, c₅ are phase shifts that can be absorbed into function arguments.
<p><strong>Step 1:</strong> Identify the form of the general solution: y = (c₁ + c₂)sin(x + c₃) + c₄e^(x+c₅)</p><p><strong>Step 2:</strong> Rewrite to expose independent constants. Let A = c₁ + c₂, then: y = A·sin(x + c₃) + c₄e^(x+c₅)</p><p><strong>Step 3:</strong> Further simplify the exponential term: c₄e^(x+c₅) = c₄·e^(c₅)·e^x = Be^x where B = c₄e^(c₅) is a single arbitrary constant</p><p><strong>Step 4:</strong> The reduced form is y = A·sin(x + c₃) + B·e^x, containing three independent arbitrary constants: A, c₃, and B</p><p><strong>Step 5:</strong> The order of the differential equation equals the number of independent arbitrary constants = 3</p><p>∴ Answer: C (Order = 3)</p>
Correct Answer: C