Give an IVP a_n(x)d^ny/dx^n + a_n-1(x)d^n-1y/dx^n-1 +...+a_1|(x)dy/dx + a_0(x)y

Question

Give an IVP a_n(x)d^ny/dx^n + a_n-1(x)d^n-1y/dx^n-1 +...+a_1|(x)dy/dx + a_0(x)y = g(x) y(x_0) = y_0, y'(x_0) = y_1, ..., y^n-1)(x_0) = y=_n-1 If the coefficients a_n(x),..., a_0(x) and the right hand side of the equation g(x) are continuous on an interval I and if a_n (x) 0 on I then the IVP has a unique solution for the point x_0 belongs to I that exists on the whole interval I. Consider the IVP on the whole real line (x^2 - 81)d^4y/dx^4 + x^4 d^3y/dx^3 + 1/x^2+81 dy/dx + y = sin(x) y(3) = 117, y'(3) = 8, y"(3) = 4, y'"(3) = 5 The Fundamental Existence Theorem for Linear Differential Equations guarantees the existence of a unique solution on the interval

Explanation / Answer

Here a0 [x] = (x2 -81) has zeros at x = -9,+9.

The other coefficients are continuous everywhere .(1/x2 +81) is continuous over R as the denominator has no real roots.

So the interval guaranteed by the existence and uniqueness theorem is (-9,+9)

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