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Solution of differential equation on interval of x values

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Determine the order of each differential equation, and whether or not the given functions are solutions of that equation on some interval of x values.
(a) (y')^2 = 4y; Y1 = X^2, Y2 = 2X^2 , Y3 = e-x
(b) Y" +9y =0; Y1 = 4sin3X, Y2 = 6sin(3X+2)

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With regard to the question of whether any given function is a solution on some interval of x values, the point is whether it's a solution on some INTERVAL of x values (and not just a solution on some set of ISOLATED points, or a solution on NO set).

Let's take a look at the given differential equations and the possible solutions to them.

(a) (y')^2 = 4y


If y_1 = x^2, then (y_1)' = 2x, so [(y_1)']^2 = (2x)^2 = 4(x^2)

and 4(y_1) = 4(x^2)

Hence [(y_1)']^2 = 4(x^2) = 4(y_1), so y_1 is a solution for ALL x; that is, y_1 a solution on the interval (-infinity, infinity).


If y_2 = 2x^2, then (y_2)' = 4x, so [(y_2)']^2 = (4x)^2 = 16(x^2)

and 4(y_2) = 4(2x^2) = 8(x^2).

Thus the question whether (y_2)' = 4(y_2) is a solution on some interval is the question ...

Solution Summary

The general meaning of a solution (of a differential equation) being a solution on an interval of x values is explained. The question of whether each of the given functions is a solution of the given equation is examined in detail. A determination of whether the function is a solution on some interval of x values is made, and the reason (as to whether/why it is a solution on an interval of x values) is explained.

Also, the orders of the given differential equations are determined and explained.