Question in Description

Question

Consider the differential equation 

Part A: Find the equation of the line tangent to the solution curve at the point (0, 4). (5 points)

Part B: Find the second derivative at (0, 4) and use it to determine the concavity of the solution curve at that point. Explain. (10 points)

Part C: Find the particular solution y = f(x) with initial condition f(0) = 4. (15 points)

Answer 1

A) Find the slope of the tangent line by evaluating dy/dx at x = 0. Use that slope and the given point, (0,4), to write the equation of the tangent line in point-slope form: y - y₁ = m (x - x₁)

B) Find d²y / dx² (i.e. the second derivative), by using product rule:

d²y / dx² = 4 [ 2(sin(x² + 2x + π/2)) + (2x + 2)cos(x² + 2x + π/2)(2x + 2) ] Evaluate at x = 0. If the second derivative is positive, the curve is concave up. If it is negative, the curve is concave down.

C) Integrate the right-hand side of the given equation by performing a u-substitution: u = (x² + 2x + π/2)

du = (2x + 2) dx so y = ∫4sinu du. Use the given point (0,4) to solve for C and arrive at the particular solution.