Finding general solution for ODE.

• Lengalicious
In summary: Sorry I typoed x's and replaced them with y's. And to solve for y as a function of x, you would use standard properties of log. Thank you for your help! In summary, the differential equation has the following solution: y(0) = 3 when x = -5, -3, and 0.
Lengalicious

Homework Statement

For the following differential equation:
1) Provide the general solution
2) Discuss for which values of x the solution is defined.
3) Find the solution of the initial value problem y(0) = 3

Homework Equations

dy/dx = (y+3)(y-5)

The Attempt at a Solution

1) so I separate variables and get (1/8)*(log(y-5)-log(y+3)) = x + C after integration.
Now i need to solve for x right? How would I go about doing this? Can't seem to figure out how to rearange.
2) I don't think i can do this until i find the general solution? But once its found do i just find its domain?
3) For this i just substitute the initial values in, solve for C and then place back in equation?

Last edited:
Lengalicious said:

Homework Statement

For the following differential equation:
1) Provide the general solution
2) Discuss for which values of x the solution is defined.
3) Find the solution of the initial value problem y(0) = 3

Homework Equations

dy/dx = (y+3)(y-5)

The Attempt at a Solution

1) so I separate variables and get (1/8)*(log(x-5)-log(x+3)) = x + C after integration.
Now i need to solve for x right? How would I go about doing this? Can't seem to figure out how to rearange.
2) I don't think i can do this until i find the general solution? But once its found do i just find its domain?
3) For this i just substitute the initial values in, solve for C and then place back in equation?

How did you get ∫dy/[(y+3)(y-5)] = (1/8)[log(x-5)-log(x-3)]?

RGV

sorry replace the x's with y's that was a typo. And i did it by splitting into partial fractions then integrating each term simply. Was that not correct?

Lengalicious said:

Homework Statement

For the following differential equation:
1) Provide the general solution
2) Discuss for which values of x the solution is defined.
3) Find the solution of the initial value problem y(0) = 3

Homework Equations

dy/dx = (y+3)(y-5)

The Attempt at a Solution

1) so I separate variables and get (1/8)*(log(y-5)-log(y+3)) = x + C after integration.
Now i need to solve for x right? How would I go about doing this? Can't seem to figure out how to rearange.
2) I don't think i can do this until i find the general solution? But once its found do i just find its domain?
3) For this i just substitute the initial values in, solve for C and then place back in equation?

You don't need to solve for x; it is already done---just read your own equation! Now, if you wanted to solve for y as a function of x, that would involve a bit more effort, but is still just elementary algebra: use standard properties of log to do it.

RGV

1. What is a general solution for an ODE?

A general solution for an ODE (ordinary differential equation) is an equation that contains all possible solutions to the differential equation. It includes a set of arbitrary constants that can be used to represent any specific solution.

2. How do you find the general solution for an ODE?

To find the general solution for an ODE, you need to solve the differential equation using various mathematical techniques such as separation of variables, integrating factors, or using a substitution. Once you have found the general solution, you can use initial conditions to determine the specific solution.

3. What is the importance of finding the general solution for an ODE?

Finding the general solution for an ODE is important because it allows us to determine all possible solutions to the differential equation. This is useful in many fields of science and engineering, such as physics, chemistry, and biology, where ODEs are commonly used to model real-world phenomena.

4. Can the general solution for an ODE always be found?

No, the general solution for an ODE cannot always be found analytically. In some cases, the differential equation may be too complex to solve, or there may not be a closed-form solution. In these cases, numerical methods or approximations can be used to find a solution.

5. How do you know if a general solution for an ODE is unique?

A general solution for an ODE is unique if it satisfies the initial conditions of the differential equation. This means that when the arbitrary constants are assigned specific values, the solution should match the given initial conditions. If there are multiple solutions that satisfy the initial conditions, then the general solution is not unique.

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