Find General & Implicit Solutions for dy/dx = e^-3y cos x (1+sin x )^2

In summary, the conversation involves finding the general solution and implicit form of an equation involving dy/dx, e^-3y, cos x, and (1+sin x)^2. The individual provides their attempt at a general solution and implicit form, but there is some confusion and it is unclear how they arrived at their equations. There is a suggestion to double check the source of the problem and follow HallsofIvy's approach.
  • #1
morbello
73
0
im asked to find the general solution to the equation below and after find the implicit form I've done some work on it and just wanted to see if I am going in the right direction.

the equation

dy[tex]/dx = e^-3y cos x (1+sin x )^2[/tex]



My attempt at a general solution


1[tex]/3 e^3 = (sin x +1)^3 = y =e^3/(sin + 1)^3 +c[/tex]

and the implicit form as

1[tex]/12 (1+sin x )^3/exp (3y)[/tex]

this is harder maths than I've done so not as sure as with most.
 
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  • #2
morbello said:
im asked to find the general solution to the equation below and after find the implicit form I've done some work on it and just wanted to see if I am going in the right direction.

the equation

dy[tex]/dx = e^-3y cos x (1+sin x )^2[/tex]
Do you mean
[tex]\frac{dy}{dx}= e^{-3y}cos x(1+ sin x)^2[/tex]?




My attempt at a general solution


1[tex]/3 e^3 = (sin x +1)^3 = y =e^3/(sin + 1)^3 +c[/tex]
Surely you don't mean this- it makes no sense. That says that y is a constant, that it is equal to (sin x+ 1)3, and that it is equal to e3 over a constant!

and the implicit form as

1[tex]/12 (1+sin x )^3/exp (3y)[/tex]

this is harder maths than I've done so not as sure as with most.
That last is not even an equation!
If you meant
[tex]\frac{dy}{dx}= e^{-3y}cos x(1+ sin x)^2[/tex]
then it separates into
[tex]e^{3y}dy= cos x (1+ sin x)^2 dx[/itex]
The left side integrates easily and the substitution u= 1+ sin x makes the right side simple.
 
  • #3
no what i ment that y was equaled to

e^3/(sinx+1)^3 +c
 
Last edited:
  • #4
morbello said:
im asked to find the general solution to the equation below and after find the implicit form I've done some work on it and just wanted to see if I am going in the right direction.

the equation

dy[tex]/dx = e^-3y cos x (1+sin x )^2[/tex]

My attempt at a general solution

1[tex]/3 e^3 = (sin x +1)^3 = y =e^3/(sin + 1)^3 +c[/tex]

The second eqn doesn't follow from the first. How did you get it? Show the calculation.

Look again at where you got the problem from. What HallsofIvy says makes the most sense.
 

1. What is the general solution for dy/dx = e^-3y cos x (1+sin x )^2?

The general solution for this differential equation is y = -1/3 ln(1+sin x) + C, where C is an arbitrary constant.

2. What is an implicit solution for dy/dx = e^-3y cos x (1+sin x )^2?

An implicit solution for this differential equation is given by the equation 3y - ln(1+sin x) = C, where C is an arbitrary constant.

3. How is the solution to dy/dx = e^-3y cos x (1+sin x )^2 affected by the value of C?

The value of C determines the specific solution to the differential equation. It shifts the graph of the solution vertically, but does not change the overall shape or behavior of the solution.

4. Can the solution to dy/dx = e^-3y cos x (1+sin x )^2 be verified?

Yes, the solution can be verified by substituting it into the original differential equation. If the resulting equation is true, then the solution is correct.

5. Are there any restrictions on the values of x and y for the solution to dy/dx = e^-3y cos x (1+sin x )^2?

Yes, the only restriction is that the natural logarithm term ln(1+sin x) must have a positive argument, which means that 1+sin x must be greater than 0. Other than that, there are no restrictions on the values of x and y for the solution.

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