PYQs Based Test - 12 : Higher order LDE

The solution of the differential equation

$\frac{d^{2}y}{d{x}^2}-\frac{dy}{dx}-2y=3e^{2x}$

Where y(0) = 0 and y’(0) = -2 is

If y = f(x) satisfies the boundary value problem y’’ + 9y = 0, y (0) = 0, y(π/2) = √2, then y(π/4) is ______

Consider the differential equation:
$2\frac{{\mathrm{d}}^2\mathrm{y}}{\mathrm{d}{\mathrm{t}}^2}+8y=0$
With initial conditions:
At t = 0, y = 0 and $\frac{\mathrm{dy}}{\mathrm{d}t}=10$ . The value of y (up to two decimal places) at t = 1 is _______.

Given that ẍ + 3x = 0, and x(0) = 1, ẋ(0) = 0, what is x(1)?

The respective expressions for complementary function and particular integral part of the solution of the differential equation $\frac{d^{4}y}{d{x}^4}+3\frac{d^{2}y}{d{x}^2}=108x^2$ are