If a function is growing or shrinking exponentially, it can be modeled using a differential equation. The equation itself is dy/dx=ky, which leads to the solution of y=ce^(kx). In the differential equation model, k is a constant that determines if the function is growing or shrinking. Be able to find roots by factoring. For a quadratic, for example, look for values of d, e, f, g such that az2 + bz + c = (d − ez)(f − gz) = 0. Then the characteristic roots are z∗ = d/e and z∗ = f/g. The number of real roots of the quadratic ax2 + bx + c = 0 is determined by the value of the discriminant d = b2 − 4ac. In this exercise, we write a function to return a value indicating the number of real roots for a quadratic equation. For the fourth order differential equation y(4) − y = 0 a friend hands us four solutions, namely, y1(x) = ex, y2(x) = e−x, y3(x) = sinh x, y4(x) = cosh x. The first and third rows in this determinant are equal, so the conclusion is W(x) =0.

RK Naresh Sir thanks for your answer.