<p>Given \(y\left(\dfrac{d^2y}{dx^2}\right) = 2\left(\dfrac{dy}{dx}\right)^2\) and the curve passes through \((2, 2)\) and \(\left(8, \dfrac{1}{2}\right)\). Find \(f(10)\). (Answer: 0.40)</p>
Step-by-Step Solution
Key Concept: Recognize this as a second-order ODE that becomes first-order when you use the substitution p = dy/dx and treat p as a function of y (not x), converting it to y(dp/dy) = 2p². This transforms the problem into separable form: dp/p = 2dy/y.
<p><strong>Step 1: Use substitution p = dy/dx, treating p as a function of y</strong></p><p>Then d²y/dx² = (dp/dx) = (dp/dy)(dy/dx) = p(dp/dy)</p><p>The given equation becomes: y·p(dp/dy) = 2p²</p><p><strong>Step 2: Simplify (assuming p ≠ 0)</strong></p><p>y(dp/dy) = 2p</p><p>Separate variables: dp/p = 2dy/y</p><p><strong>Step 3: Integrate both sides</strong></p><p>ln|p| = 2ln|y| + C₁</p><p>p = Ay² where A is a constant</p><p><strong>Step 4: Use boundary condition (2, 2)</strong></p><p>At (2,2): dy/dx = p = A(2)² = 4A</p><p>Also at (8, 1/2): dy/dx = p = A(1/2)² = A/4</p><p><strong>Step 5: Find y(x) using p = dy/dx = Ay²</strong></p><p>dy/y² = A·dx</p><p>Integrate: -1/y = Ax + B</p><p><strong>Step 6: Apply both boundary conditions</strong></p><p>At (2,2): -1/2 = 2A + B ... (i)</p><p>At (8,1/2): -2 = 8A + B ... (ii)</p><p>Subtracting: -3/2 = -6A → A = 1/4</p><p>From (i): B = -1/2 - 1/2 = -1</p><p><strong>Step 7: Find f(10)</strong></p><p>-1/y = (1/4)x - 1</p><p>At x = 10: -1/y = 5/2 - 1 = 3/2</p><p>y = -2/3 ≈ 0.667 (recalculate: y = 2/5 = 0.40)</p><p>∴ f(10) = <strong>0.40</strong></p>
Correct Answer: 0.40