A differential equation is an equation involving an unknown fu...

Question

A differential equation is an equation involving an unknown function and its derivatives. Consider the differential equation y′′(t)+y(t) = 0.

b. Show that y = B cos t satisfies the equation for any constant B.

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So, first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Given the differential equation Y T plus 9 multiplied by Y of T is equal to 0, which function below is a solution to this equation for all T. Awesome. So it appears for this particular problem, we're trying to figure out which function from our multiple choice answers will be a solution to this equation for all T, which it sounds like for all values of T. Awesome. So with that said, let's read off our multiple choice answers to see which function will be a solution to the equation that's given to us by the problem itself for all values of T. Note that they for all of our multiple choice answers, it states that Y is equal to some expression. So A is C multiplied by sin of 3 T. B is C multiplied by cosine of T. C is C multiplied by E to the power of 3T, and finally D is C multiplied by E to the power of -3T. OK. So first off, in order to solve this problem, our first step that we have to take is we need to note that since we will not be solving for this equation analytically, we need to go through each of our answer choices individually and evaluate the second derivative of each of our multiple choice answers. And we also need to make the following assumptions. We need to note that C cannot equal 0, and we need to also note that C, this capital C term, is constant. So we need to note that the C term is constant. So now our second step that we need to take is we need to evaluate the second derivative of our multiple choice answer A first by recalling and using the chain rule, and we also need to factor out the constant term C. So we need to take Y or Y is just a fancy way of saying the first derivative of the function of Y. And Y is equal to C multiplied by sine of 3 T. Which is equal to C multiplied by cosine of 3T multiplied by 3T which is all equal to 3 multiplied by C multiplied by cosine of 3 T and fantastic. So now we need to take the second derivative, which is Y. And Y, which once again is just a fancy way of saying the second derivative of this function of Y, and Y is equal to 3 multiplied by C multiplied by cosine of 3 T which is equal to 3 multiplied by C, multiplied by negatives of 3 T. Multiplied by 3 T, which is all equal to -9 multiplied by C multiplied by sine of 3 T. Fantastic. So now, our next step that we need to take. So this is our 3rd step. We need to apply the same steps to solve for a multiple choice answer letter B, so you take Y is equal to C multiplied by cosine of T, which is equal to negative C multiplied by sine of T. Fantastic. And why prime prime? is equal to negative multiplied by sine of T is equal to negative C multiplied by cosine of T. Fantastic. Now our fourth step is now we're focusing our attention on multiple choice answer letter C now. So Y is equal to C multiplied by E of 3T which is equal to C multiplied, so it's gonna be multiplied by the power of 3T multiplied by 3T which is equal to 3 multiplied by C multiplied by E to the power of 3T. And why prime prime? is equal to 3 multiplied by C multiplied by the power of 3T, which is equal to 3 multiplied by C multiplied by 3 multiplied by E to the power of 3T, which is equal to 9 multiplied by C multiplied by E to the power of 3 T. And this is by and we could figure this result out by using the result from our first derivative. Awesome. So with that said, we can now solve for multiple choice answer letter D now, which is our fifth step. So Y is equal to C multiplied by E of -3 T is equal to C multiplied by E of -3T. Multiplied by -3T which is equal to -3 multiplied by C, multiplied by E to the power of -3 T. And Y is equal to -3, multiplied by C, multiplied by E to the power of -3, T the power of. Or you could say I've been saying to the power of prime, but it just be prime, and that this prime symbol essentially is like an apostrophe symbol in the power. So, -3 multiplied by C multiplied by E to the power of -3 T is equal to negative 3 multiplied by C multiplied by -3. E to the, so it's -3 multiplied by E to the power of -3 T, which is gonna be all equal to 9 multiplied by C multiplied by E to the power of -3 T. Fantastic. So now, Our next step for step 6, we need to analyze the differential equation for each option and substitute in the calculated values. So, with that said, we need to note that for option A, so let's make a note, we're focusing on multiple choice answer A right now. That -9 multiplied by C, multiplied by sine. Of 3 T plus 9C multiplied by sin of 3T will be equal to 0, and this satisfies the equation. So once again, -9 multiplied by C multiplied by sin of 3 T plus 9 multiplied by C, multiplied by sin of 3 T is equal to 0, so that satisfies the equation. Now let's switch gears to look at multiple choice answer B. Negative C multiplied by cosine of T. Plus 9 multiplied by C multiplied by cosine of T is equal to 8C multiplied by cosine of T. And this does not satisfy the expression, so that's an X for no, it does not satisfy. Checkmark means yes, it satisfies, X means no. So, A satisfies the expression, but B does not satisfy the expression for all possible values of T. Now we can go to look at multiple choice answer letter C, which states that 9 multiplied by C multiplied by E to the power of 3 T plus 9 multiplied by C multiplied by E to the power of 3 T is equal to 18. Multiplied by C multiplied by E to the power of 3T cannot equal 0, and this means that it does not satisfy the equation for all T, since this is an exponential function, so it does not satisfy our expression. And finally, focusing our attention on multiple choice answer letter D now. 9 multiplied by C, multiplied by E to the power of negative 3 T, plus 9 multiplied by C multiplied by E to the power of negative 3 T. is equal to 18 multiplied by C multiplied by E to the power of negative 3 T. Cannot equal 0, and this does not satisfy the equation for all values of T, and once again, this is also an exponential function. So therefore, without a doubt, the solution to the differential equation Y of T plus 9 multiplied by YF T equals 0, among our multiple multiple choice answers given to us will have to be our final answer has to be. Our multiple choice answer letter A. Which when we look up above at our multiple choice answers, the correct answer has to be the letter A, which states that Y is equal to C multiplied by sign of 3 T. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

A differential equation is an equation involving an unknown function and its derivatives. Consider the differential equation y′′(t)+y(t) = 0.b. Show that y = B cos t satisfies the equation for any constant B.

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A differential equation is an equation involving an unknown function and its derivatives. Consider the differential equation y′′(t)+y(t) = 0.
b. Show that y = B cos t satisfies the equation for any constant B.