Problem 9 - Entrance Test

y = e^(ax) cos(bx). dy/dx = ae^(ax) cos(bx) - be^(ax) sin(bx). d²y/dx² = a²e^(ax) cos(bx) - abe^(ax) sin(bx) - abe^(ax) sin(bx) - b²e^(ax) cos(bx) = (a² - b²)e^(ax) cos(bx) - 2abe^(ax) sin(bx) = (a² - b²)y - 2ab(e^(ax) sin(bx)). Now, dy/dx = ay - be^(ax) sin(bx), so be^(ax) sin(bx) = ay - dy/dx. Substituting this into the expression for d²y/dx², we get d²y/dx² = (a² - b²)y - 2ab(ay - dy/dx)/b = (a²-b²)y -2a(ay-dy/dx) = (a²-b²)y - 2a²y + 2a dy/dx. Not among solutions. We know dy/dx = ay - be^(ax)sin(bx) => dy/dx - ay = -be^(ax)sin(bx) => e^(ax)sin(bx) = (ay - dy/dx)/b. d²y/dx² = (a² - b²)y - 2ab((ay - dy/dx)/b) = (a² - b²)y - 2a(ay - dy/dx) = (a² - b²)y - 2a²y + 2a(dy/dx). This is still wrong. From dy/dx = ae^(ax)cos(bx) - be^(ax)sin(bx), d²y/dx² = (a²-b²)e^(ax)cos(bx) - 2abe^(ax)sin(bx). (a² + b²)y = (a² + b²)e^(ax)cos(bx). Therefore, d²y/dx² = (a²-b²)e^(ax)cos(bx) - 2abe^(ax)sin(bx)= (a² + b²) e^(ax)cos(bx) -2b²e^(ax)cos(bx)- 2abe^(ax)sin(bx).