# Solving ODE via Separation Method: Need Assistance

• Mugged
Cmultiply both sides by -1e^(-y) = -e^x - Ctake natural log of both sides-y = ln(-e^x - C)solve for yy = -ln(-e^x - C)This general solution exists only for C > e^x or x < ln(C). In summary, the discussion is about solving a given ODE, which involves finding a general solution. One person suggests using a separation method, but is having trouble applying the natural log. Another person offers a solution by moving the minus sign to the other side of the equation and remembering to include the constant of integration. The final solution
Mugged
Can anyone help me solve this ODE:
..in other words, find a general solution?

dy/dx = e^(x+y)

I use a separation method, but i can't take the natural log of -e^(-y).
So, help?

Perhaps you can move the minus in front of e-y to the other side of your equation before you apply log.

Looks to me like you have forgotten the constant of integration. You should get something like $y= -ln(C- e^x)$ which exists only for $C> e^x$ or $x< ln(C)$

Mugged said:
Can anyone help me solve this ODE:
..in other words, find a general solution?

dy/dx = e^(x+y)

I use a separation method, but i can't take the natural log of -e^(-y).
So, help?

e^(x+y) = e^x * e*y
e^(-y) dy = e^x dx
integrate both sides

Sure, I would be happy to assist you with solving this ODE using the separation method. First, let's review the steps for solving an ODE using separation:

1. Rewrite the ODE in the form dy/dx = f(x,y)
2. Separate the variables by moving all terms containing y to one side and all terms containing x to the other side.
3. Integrate both sides of the equation.
4. Solve for y to find the general solution.

Applying these steps to the given ODE, we have:

1. dy/dx = e^(x+y)
2. dy/e^y = e^x dx
3. Integrate both sides:
∫dy/e^y = ∫e^x dx
Using the substitution u = -y, du = -dy, this becomes:
-∫du/e^u = ∫e^x dx
-1/e^u = e^x + C
4. Solve for y:
-1 = e^(x-y) + Ce^y
Rearranging, we get:
e^(x-y) = Ce^y - 1
Taking the natural log of both sides:
x-y = ln(Ce^y - 1)
Simplifying:
y = x - ln(Ce^y - 1)
This is the general solution to the given ODE.

I hope this helps and feel free to ask for further clarification if needed.

## 1. What is the separation method for solving ODEs?

The separation method is a technique used to solve ordinary differential equations (ODEs) by separating the variables in the equation and then integrating each side separately.

## 2. How does the separation method work?

The separation method works by rearranging the ODE so that all terms containing the dependent variable are on one side and all other terms are on the other side. Then, the equation is integrated with respect to the dependent variable to obtain a solution.

## 3. When is the separation method useful for solving ODEs?

The separation method is useful for solving ODEs when the equation can be separated into two parts, one with only the dependent variable and its derivatives, and the other with only the independent variable. This method is particularly useful for first-order ODEs.

## 4. Are there any limitations to using the separation method for solving ODEs?

Yes, the separation method can only be used for certain types of ODEs that can be separated into two parts. Additionally, it may not always be possible to obtain an analytical solution using this method, and numerical methods may be required instead.

## 5. What are some tips for using the separation method to solve ODEs?

Some tips for using the separation method include: identifying when the method can be applied, checking for any initial or boundary conditions, being careful with integrating factors and constants, and double-checking the solution for accuracy.

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