Differential Equations
Higher order ODEs — constant coefficients
Grade Class 12
Question:
<p>The solution of \(\dfrac{d^2y}{dx^2} - 5\dfrac{dy}{dx} + 6y = 0\) is:</p>
<span>\(y=C_1e^{2x}+C_2e^{3x}\)</span>
<span>\(y=C_1e^{-2x}+C_2e^{-3x}\)</span>
<span>\(y=C_1e^{2x}+C_2e^{-3x}\)</span>
<span>\(y=(C_1+C_2x)e^{3x}\)</span>
Step-by-Step Solution
Key Concept: Characteristic equation: m^2 - 5m + 6 = 0.
<div class='solution'><p><strong>Step 1:</strong> Characteristic equation: $m^2 - 5m + 6 = 0$.</p>
<p>$(m-2)(m-3) = 0 \implies m = 2, 3$. Two distinct real roots.</p>
<p><strong>Step 2:</strong> General solution: $y = C_1 e^{2x} + C_2 e^{3x}$. <strong>Answer: (A)</strong>.</p>
<p class='key-concept'>🔑 Key Concept: For $ay'' + by' + cy = 0$: solve $am^2+bm+c=0$. Distinct roots $m_1, m_2$ → $y = C_1e^{m_1x}+C_2e^{m_2x}$. Equal roots → $(C_1+C_2x)e^{mx}$.</p></div>
Correct Answer: 1