Question

Consider the linear second-order differential equation:

d^2y/ dx^2 + a1(x)dy/dx +ao(x)y(x)=0

Note that this equation is linear because y(x) and its derivatives appear only to the first power and there are no cross terms. It does not have constant coefficients, however, and there is no general, simple method for solving it like there is if the coefficients were constants. In fact, each equation of this type must be treated more or less individually. Nevertheless, because it is linear, we must have that if y1(x) and y2(x) are any two solutions, then a linear combination.

y(x)=c1y1(x)+c2y2(x) where c1 and c2 are constants, is also a solution. Prove that y(x) is a solution.

Answer 1

To prove that the linear combination y(x) = c1y1(x) + c2y2(x) is also a solution to the given linear second-order differential equation, we need to substitute it into the equation and show that it satisfies the equation. By substituting y(x) = c1y1(x) + c2y2(x) into the differential equation, we can simplify and demonstrate that the left-hand side of the equation becomes 0, proving that y(x) = c1y1(x) + c2y2(x) is a solution.

Step-by-step explanation:

In order to prove that the linear combination y(x) = c1y1(x) + c2y2(x) is also a solution to the given linear second-order differential equation, we need to substitute it into the equation and show that it satisfies the equation. Let's substitute y(x) = c1y1(x) + c2y2(x) into the differential equation and simplify:

d^2y/dx^2 + a1(x)dy/dx + a0(x)y = 0

d^2(c1y1(x) + c2y2(x))/dx^2 + a1(x)d(c1y1(x) + c2y2(x))/dx + a0(x)(c1y1(x) + c2y2(x)) = 0

c1(d^2y1/dx^2 + a1(x)dy1/dx + a0(x)y1) + c2(d^2y2/dx^2 + a1(x)dy2/dx + a0(x)y2) = 0

Since both y1(x) and y2(x) are solutions to the differential equation, we know that d^2y1/dx^2 + a1(x)dy1/dx + a0(x)y1 = 0 and d^2y2/dx^2 + a1(x)dy2/dx + a0(x)y2 = 0. Therefore, when we substitute y(x) = c1y1(x) + c2y2(x), the left-hand side of the equation becomes c1(0) + c2(0) = 0. This means that y(x) = c1y1(x) + c2y2(x) is a solution to the given linear second-order differential equation.