7–16. Verifying general solutions Verify that the given functi...

Question

7–16. Verifying general solutions Verify that the given function is a solution of the differential equation that follows it. Assume C, C1, C2 and C3 are arbitrary constants.

y(t) = C₁ sin4t + C₂ cos4t; y''(t) + 16y(t) = 0

Accepted Answer

Welcome back, everyone. A F of X equals m of 2 X plus M cosine of 2 X, a solution to the differential equation F X plus 4 F of X equals 0, where M and N are arbitrary constants a yes, and B, no. For this problem, let's focus on the differential equation. We want to identify the second derivative. So let's begin by finding the first derivative, right? We want to differentiate M multiplied by sine of 2 X plus M multiplied by cosine of 2 X, where M and N are constants. So starting with the first term, we can factor out M, the derivative of sine is cosine. So we get cosine of 2 X multiplied by 2 according to the chain rule because we have a composite function. So we get 2M cosine of 2 X, that's our derivative. And now considering our next term, we get plus M, the derivative of cosine is negative sine. So we can multiply by negative sign of 2 X, multiplied by the derivative of 2 X, which is again 2, right? So we can write minus. 2 and sign of 2 X. That's our first derivative. Now, because we need to identify the second derivative, we're going to differentiate this result, that's going to be F of X. We are going to begin with the derivative of 2m cosine of 2 X and subtract the derivative of 2 M sine of 2 X, right? Starting with the first derivative, we get 2 M, that's our constant multiplied by the derivative of cosine is negative signs, we get negative sine of 2 X multiplied by the derivative of 2 X, which is 2, right? And then we're subtracting to an. The derivative of sin of 2 X is cosine of 2 X multiplied by 2, right? Simplifying, we get -4 M. Sign of 2X minus 4 and cosine of 2 X. Now, let's substitute this expression into the differential equation. We get -4M of 2 X minus 4M cosine of 2 X, and we're adding 4 F of X, right? So let's add 4 and min of 2 X plus N. Cosine of 2 X. Let's factor out -4. We get -4 M. Sign of 2 X plus M cosine of 2 X plus 4 impreis M of sin 2 X plus M cosine of 2 X. We can now show that this is 0, right? Because we have the same factor and -4 + 4 is 0. So we get 0 equals 0, meaning that F of X is a solution, and the correct answer to this problem is A. Thank you for watching.

7–16. Verifying general solutions Verify that the given function is a solution of the differential equation that follows it. Assume C, C1, C2 and C3 are arbitrary constants.
y(t) = C₁ sin4t + C₂ cos4t; y''(t) + 16y(t) = 0

Key Concepts

General Solution of a Differential Equation

Verification by Substitution

Second Derivative and Trigonometric Functions

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