Differiental Equation, One Equation, Three Answers

[cmhex]

EDIT: The confusion here was due to a problem with the quiz the questions came from. I'm cleaning up this post to reflect this because I don't want somebody to see this post and get confused.

7y" + 7y = 0

Valid Solution, Solve the Equation, Solve the first order linear differential

C1 cos x + C2 sin x is a valid answer for all three parts.

EDIT: The response below reflect are in response to the old invalid information post, they may have perfectly good and valid information in them however.

 

[stapel]

7y''+7y = 0

First, it asks for a valid solution. No problem

C1 cos x + C2 sin x

Does the above perhaps mean "y=" the stated expression?

...solve the equation

C1 e^-x cos x + C2 e^-x sin of x
e^-3x + e^x

Are there maybe "y=" somewhere in the exercise statements? Because, as posted, neither of these is an "equation".

Note: There is a difference between finding "a" solution to a generic equation (such as when "y" is not specified) and finding "the" solution to a specific equation (where "y" is specified). Is this what you're asking about?

 

[HallsofIvy]

First the equation in question

7y''+7y = 0, used in all three questions.

First, it asks for a valid solution. No problem

C1 cos x + C2 sin x

Okay, y(x)= that is the "general" solution to the given differential equation

Second, it asks to solve the equation

C1 e^-x cos x + C2 e^-x sin of x

No, it doesn't. First, that is not an equation and, second, if it were given equal to something, what are we to "solve" for? That looks like (if we were given "y= " that) the solution to a different differential equation.

Thirdly, Solve the first order linear differential equation

e^-3x + e^x

Again, that is NOT an "equation"of any kind and certainly is not a differential equation.

My first question is obviously related to a missing concept, I'm not understanding the difference between a valid solution and solving the problem because this sounds like the same thing to me. I included the third part mostly out of curiosity, I knew what was being asked and how to get it, I'm just interested to know (if it can be explained easily) what they were getting at by putting these three questions together.

You seem to have difficulty reading the problems. Perhaps if you just copied the problems themselves here we could help you with that. The last to look like they might be asking you to find the differential equation that these functions satisfy.

 

[cmhex]

After asking the teacher about it it seems there's some sort of problem with the quiz. Which makes sense to me after today's lecture. All of the answers would be that 'first' answer I provided.

 

[HallsofIvy]

First the equation in question

7y''+7y = 0, used in all three questions.

First, it asks for a valid solution. No problem

C1 cos x + C2 sin x

Second, it asks to solve the equation

C1 e^-x cos x + C2 e^-x sin of x

\(\displaystyle y(x)= e^{-x}cos(x)+ C_2e^{-x}sin(x)\) is the general solution to the differential equation
\(\displaystyle y''+ 2y'+ 2y= 0\)

Thirdly, Solve the first order linear differential equation

e^-3x + e^x

\(\displaystyle y(x)= C_1e^{-3x}+C_2e^{x}\) is the general solution to the differential equation
\(\displaystyle y''+ 2y'- 3y= 0\)

The specific function \(\displaystyle y(x)= e^{-3x}+ e^x\) satisfies that differential equation and the conditions y(0)= 2, y'(0)= -2

My first question is obviously related to a missing concept, I'm not understanding the difference between a valid solution and solving the problem because this sounds like the same thing to me. I included the third part mostly out of curiosity, I knew what was being asked and how to get it, I'm just interested to know (if it can be explained easily) what they were getting at by putting these three questions together.