Is this general solution for ODE correct?

Based on the conversation, the general solution for the given ODE is x = (Ae^(sin t))/3 and the solution provided by the expert is correct. In summary, the general solution for the ODE dx/dt = 3x^(2) cos t is x = (Ae^(sin t))/3, and the expert has provided a correct solution for making x the subject of the solution.
  • #1
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Find the general solution of the following ODE:

dx/dt = 3x^(2) cos t



Make x the subject of the solution.



Heres my solution, is this correct?

dx/dt = 3x^(2) cos t

dx/3x^(2) = cos t dt

Integrating both sides gives:

ln (3x^(2)) = sin t + C

3x^(2) = e^(sin t + C)

3x^(2) = Ae^(sin t)

x^(2) = (Ae^(sin t))/3)

x = SQRT(Ae^(sin t))/3)
 
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  • #2
Jamiey1988 said:
Heres my solution, is this correct?

dx/dt = 3x^(2) cos t

dx/3x^(2) = cos t dt

Integrating both sides gives:

ln (3x^(2)) = sin t + C


here is where you went wrong,

1/3x2 can be written as x-2/3.

You know that ∫xn dx = xn+1/(n+1) + C for n≠-1
 
  • #3
So from what you have said:

∫dx/3x^(2) = -1/3x + C or (-x^(-1)/3) + C

Giving a solution of:

-1/3x + C = sin t
 
  • #4
That should be correct.
 
  • #5
You can easily check your answer by plugging it back into the original differential equation and seeing if it works.
 

1. What is a general solution for an ODE?

A general solution for an ODE (ordinary differential equation) is a function that satisfies the differential equation for all possible values of the independent variable. It contains a constant of integration, which allows for an infinite number of possible solutions.

2. How can I check if a general solution for an ODE is correct?

To check if a general solution for an ODE is correct, you can substitute the function into the original differential equation and see if it satisfies the equation for all values of the independent variable. Additionally, you can also take the derivative of the function and verify that it matches the given derivative in the differential equation.

3. What is the difference between a general solution and a particular solution for an ODE?

A general solution is a function that satisfies the differential equation for all possible values of the independent variable, while a particular solution is a specific function that satisfies the equation for a given set of initial conditions. A particular solution can be obtained from a general solution by substituting the initial conditions into the function and solving for the constant of integration.

4. Can there be more than one general solution for an ODE?

Yes, there can be an infinite number of general solutions for an ODE. This is because a general solution contains a constant of integration, which can take on different values and result in different functions that still satisfy the differential equation.

5. Is a general solution for an ODE always unique?

No, a general solution is not always unique. As mentioned before, it can have an infinite number of solutions due to the constant of integration. Additionally, for certain types of differential equations, there may be multiple general solutions that satisfy the equation for different ranges of the independent variable.

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