Differential Equations
Higher order differential equations reduced to first order
Grade 12
Question:
<p>Use the methods of solving first order differential equation to find the general solution of \(\frac{d^2y}{dx^2} = 3\left(\frac{dy}{dx}\right)^{3/2}\).</p>
<p>\(\frac{-1}{p^3} = ax + b\)</p>
<p>\(\frac{1}{p^3} = ax + b\)</p>
<p>\(p^3 = ax + b\)</p>
<p>\(p^2 = ax + b\)</p>
Step-by-Step Solution
Key Concept: Recognize this as a second-order equation without explicit y-dependence. Substitute p = dy/dx to reduce it to a first-order separable equation in p and x, then integrate twice.
<p><strong>Step 1:</strong> Let p = dy/dx, so d²y/dx² = dp/dx</p><p>The equation becomes: dp/dx = 3p^(3/2)</p><p><strong>Step 2:</strong> Separate variables: dp/p^(3/2) = 3dx</p><p><strong>Step 3:</strong> Integrate both sides: ∫p^(-3/2)dp = ∫3dx</p><p>-2p^(-1/2) = 3x + C₁</p><p>Or: -2/√p = 3x + C₁</p><p><strong>Step 4:</strong> Solve for p: √p = -2/(3x + C₁)</p><p>p = 4/(3x + C₁)²</p><p><strong>Step 5:</strong> Since p = dy/dx: dy/dx = 4/(3x + C₁)²</p><p><strong>Step 6:</strong> Integrate again: y = ∫4(3x + C₁)^(-2)dx = -4(3x + C₁)^(-1)/3 + C₂</p><p>∴ Answer: y = -4/[3(3x + C₁)] + C₂ or equivalent form</p>
Correct Answer: A