CLEP Calculus Test: Deriving General Solution to y' = sin X

[3.141592654]

Homework Statement

I was looking into taking the Calculus CLEP Exam and came across this practice problem:

Derive the general solution to the following differential equation: y' = sin x.

A. sin y + cos x = C
B. tan y = C
C. sin y - cos x = C
D. cos x = C
E. cos x - sin y = C

The Attempt at a Solution

The first thing I thought was that the question was asking for me to integrate y'=sin X. If y' = sin X, then y = -cos x + C. I didn't know what they wanted me to do so I simply set this equal to zero, so -cos x + C = 0 and therefore cos X=C. However, D is not the correct answer. So, my question is: what is it that the question is asking me to do when it says it wants a "general solution to ...[a] differential equation"?

 

[3.141592654]

It might be relevant to state that choices A-E are answers for a multiple choice problem and not five parts to the problem.

 

[Mark44]

Homework Statement

I was looking into taking the Calculus CLEP Exam and came across this practice problem:

Derive the general solution to the following differential equation: y' = sin x.

A. sin y + cos x = C
B. tan y = C
C. sin y - cos x = C
D. cos x = C
E. cos x - sin y = C

The Attempt at a Solution

The first thing I thought was that the question was asking for me to integrate y'=sin X. If y' = sin X, then y = -cos x + C. I didn't know what they wanted me to do so I simply set this equal to zero, so -cos x + C = 0 and therefore cos X=C. However, D is not the correct answer. So, my question is: what is it that the question is asking me to do when it says it wants a "general solution to ...[a] differential equation"?

Your first inclination was correct. The solution to this differential equation is y = -cos x + C. That is the general solution. I have no idea why this is not listed as an option. Is what you showed the exact wording of the problem?

 

[3.141592654]

Yea, the text I provided is the exact wording. The answer, which was listed at the bottom of the page, was listed as (A) and states:

A. In order to be correct, a general solution to a differential equation must satisfy both the homogenous and non-homogenous equations.

 

[Mark44]

Our solution satisfies the DE. If y = -cos x + C, then dy/dx = sin x.

The homogeneous problem would be y' = 0, for which the solution is y = C. The nonhomogeneous problem is y' = sin x, for which the general solution is y = -cos x + C.

I think there might be a typo in the CLEP answer. They might have meant to say y + cos x = C, which would be equivalent to our answer, instead of sin y + cos x = C.

 

[3.141592654]

Can you explain what the homogeneous and nonhomogeneous problems are? I haven't come across those terms before.

 

[Mark44]

A differential equation can be represented as f(y, y', y'', ..., y⁽ⁿ⁾) = g(x). On the left side you have some function of y and its derivatives. On the right side there is some other function of the independent variable.

The homogeneous equation is f(y, y', y'', ..., y⁽ⁿ⁾) = 0. The one above is the nonhomogeneous equation. For your problem, you have y' = sin x. (There is no y term.) The homogeneous equation is y' = 0, and the nonhomogeneous equation is y' = sin x.

Another example is y'' + y = x, a nonhomogeneous DE. The related homogeneous equation is y'' + y = 0, whose solution is y = Asin x + Bcos x. The general solution of the nonhomogeneous equation turns out to be y = Asin x + Bcos x + x, and is made up of the solution to the homogeneous equation plus what is called a particular solution to the nonhomogeneous problem.

The term "homogeneous" is also used in a different context in differential equations, which can lead to some confusion. In that sense of the word, a differential equation in the form y' = f(y/x) is said to be homogeneous. My sense is that in the world of differential equations, the description I gave before is much more commonly used.