Solving Separable Differential Equation with Initial Condition

In summary, a separable differential equation is a type of differential equation where the variables can be separated into different functions, making it possible to solve the equation by integrating each side separately. To solve a separable differential equation with initial condition, the variables must be separated and then integrated, using the initial condition to find the constant of integration. However, not all differential equations can be solved using separation of variables, as this method only works for separable equations. Initial conditions are important in solving differential equations as they provide a starting point and help determine the constant of integration. There are limitations to using separation of variables, such as not all equations being separable and the integration process becoming too complex for analytical solutions.
  • #1
alchal
3
0
QUESTION:

Solve the separable differential equation
dy/dx = sqrt(4y+64), Initial Condition: y(4)=9,
and find the particular solution satisfying the initial condition.

MY ATTEMPT:

(dy/dx)^2 = 4y+64
((dy/dx)^2)-4y = 64
,/' (((dy/dx)^2)-4y) dx = ,/' 64 dx

Is this the right method? If so, not sure how to integrate (dy/dx)^2

Any suggestions?
 
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  • #2
dy/sqrt(4y+64) = dx
 
  • #3
Thank you!
 

1. What is a separable differential equation?

A separable differential equation is a type of differential equation where the variables can be separated into different functions that only depend on one variable each. This makes it possible to solve the equation by integrating both sides separately.

2. How do you solve a separable differential equation with initial condition?

To solve a separable differential equation with initial condition, you first need to separate the variables and then integrate both sides. After integrating, you can use the initial condition to find the constant of integration and obtain the final solution.

3. Can all differential equations be solved using separation of variables?

No, not all differential equations can be solved using separation of variables. This method only works for equations that are separable, meaning the variables can be separated into different functions.

4. What is the importance of initial conditions in solving differential equations?

Initial conditions are important in solving differential equations because they provide a starting point for the solution and help determine the constant of integration. Without initial conditions, the solution may have multiple possible solutions.

5. Are there any limitations to using separation of variables to solve differential equations?

Yes, there are limitations to using separation of variables to solve differential equations. This method only works for equations that are separable, and some equations may not have a separable form. Additionally, even if an equation is separable, the integration process may become too complex to solve analytically.

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