Question

Answer Problems 1-10 without referring back to the text. Fill in the blank or answer true or false. If y = sin5x is a solution of a homogeneous linear second-order differential with constant coefficients, then the general solution of the DE is_____

Short answer

y = c₁cos5x + c₂ sin5x

Step-by-step solution

Definition

The general solution of the differential equation is the correlation between the variables x and y, which is received after removing the derivatives (i.e., integration) where the relation includes arbitrary constants to represent the order of an equation.

General solution

Given $y=\sin 5x$is a solution of the homogeneous linear second-order differential equation,

The auxiliary roots must be complex and are of the form m = 0+5i

The 0 comes from the fact that there is no exponential term so that term must take the form of $e^{0x}$.

If the auxiliary root is complex, then the general solution takes the form of the equation$y=c_1\cos 5x+c_2\sin 5x$