<p>The order of differential equation whose general solution is given by \(y = (c_1 + c_2)\sin(x + c_3) - c_4 e^{x+c_5}\) 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 constants c₁, c₂, c₃, c₄, c₅ that cannot be reduced or combined into fewer independent parameters.
<p><strong>Step 1:</strong> Identify all arbitrary constants in the general solution:</p><p>y = (c₁ + c₂)sin(x + c₃) - c₄e^(x+c₅)</p><p><strong>Step 2:</strong> Determine independent arbitrary constants by analyzing their roles:</p><ul><li>(c₁ + c₂) acts as a single parameter (let's call it C₁)</li><li>c₃ is independent (phase shift)</li><li>c₄ is independent (amplitude of exponential term)</li><li>c₅ appears in the exponent as e^(x+c₅) = e^x · e^(c₅), which is equivalent to c₄e^(x+c₅) = (c₄e^(c₅))e^x, so c₅ can be absorbed into c₄</li></ul><p><strong>Step 3:</strong> Count truly independent constants:</p><p>We have exactly 3 independent arbitrary constants:</p><ul><li>C₁ = c₁ + c₂ (one constant)</li><li>c₃ (one constant)</li><li>Combined effect of c₄ and c₅ (one constant, since e^(c₅) can be absorbed into c₄)</li></ul><p><strong>Step 4:</strong> Alternatively, rewrite more clearly:</p><p>y = Asin(x + c₃) - Be^x, where A = c₁ + c₂ and B = c₄e^(c₅)</p><p>This has exactly 3 independent arbitrary constants: A, c₃, and B.</p><p><strong>∴ Answer: 3</strong></p>
Correct Answer: 3