Question

Find the general solution of the following homogeneous second-order linear differential equation. Write the final answer in the box.

(ii) ( \frac{d^{2} y}{d x^{2}} + 1 = 0 ** Please provide the step-by-step process for finding the general solution to this differential equation.

Answer 1

The general solution to the second-order linear homogeneous differential equation d²y/dx² + 1 = 0 is found by integrating twice and is y(x) = -(x²/2) + C₁x + C₂, where C₁ and C₂ are constants of integration.

Step-by-step explanation:

Step-by-Step Solution of a Differential Equation

The differential equation given is d2y/dx2 + 1 = 0. To solve this second-order linear homogeneous differential equation, we look for solutions that satisfy the equation. The first step is to rewrite the equation as d2y/dx2 = -1, indicating that the second derivative of y with respect to x is equal to -1.

Now, let's integrate the equation with respect to x to get the first derivative of y. We find that dy/dx = -x + C1, where C1 is the constant of integration. Integrating again to find y, we have y = -(x2/2) + C1x + C2, where C2 is another constant of integration.

Therefore, the general solution to the differential equation is y(x) = -(x2/2) + C1x + C2, where C1 and C2 are arbitrary constants.