Differential Equations
Formation of Differential Equations
Grade None
Question:
<p>We have <br/> \( y = c_1 e^{c_2 x} \)<br/> Which of the following is the differential equation satisfied by this?</p>
<p>\( yy'' = (y')^2 \)</p>
<p>\( yy'' = y' \)</p>
<p>\( y'' = (y')^2 \)</p>
<p>\( yy'' + (y')^2 = 0 \)</p>
Step-by-Step Solution
Key Concept: Eliminate the two arbitrary constants c₁ and c₂ by differentiating twice and creating two equations to relate y, dy/dx, and d²y/dx² without constants.
<p><strong>Step 1:</strong> Given: y = c₁e^(c₂x)</p><p><strong>Step 2:</strong> Differentiate once: dy/dx = c₁c₂e^(c₂x) = c₂y</p><p><strong>Step 3:</strong> Differentiate again: d²y/dx² = c₂(dy/dx) = c₂·c₂y = c₂²y</p><p><strong>Step 4:</strong> From Step 2: c₂ = (1/y)(dy/dx)</p><p><strong>Step 5:</strong> Substitute into Step 3: d²y/dx² = [(1/y)(dy/dx)]² · y = (dy/dx)²/y</p><p><strong>Step 6:</strong> Rearrange: y(d²y/dx²) = (dy/dx)²</p><p>∴ Answer: y·y'' = (y')² or y·d²y/dx² - (dy/dx)² = 0</p>
Correct Answer: A