General solution of a differential equation

In summary, the conversation is about finding the general solution to a differential equation in implicit form. The equation involves separating variables and manipulating the expression to get it into the form dy/dx = f(x)g(y). The person is asking for assistance in understanding the algebra involved in solving the equation.
  • #1
Briggs
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Homework Statement


Find the general solution to the differential equation in implicit form.
http://www.texify.com/img/%5Clarge%5C%21%5Cfrac%7Bdy%7D%7Bdx%7D%3D%28cosx-sinx%29e%5E%7Bcosx%2Bsinx%2By%7D.gif

Homework Equations


http://www.texify.com/img/%5Clarge%5C%21%5Cfrac%7Bdy%7D%7Bdx%7D%3Df%28x%29g%28y%29.gif

The Attempt at a Solution


Am I correct in assuming that this is a separation of variables problem? I can't seem to grasp the algebra to move the y to the left hand side. Could anybody give me a nudge in the right direction?
 
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  • #2
split exp{cosx+sinx+y} into exp(cosx+sinx)*exp(y)
 
  • #3



Yes, this is a separation of variables problem. To solve it, you need to separate the variables x and y onto opposite sides of the equation. This can be done by multiplying both sides by dx and dividing both sides by (cosx-sinx)e^(cosx+sinx+y). This will give you the following equation:

dy/(cosx-sinx)e^(cosx+sinx+y) = dx

Now, you can integrate both sides with respect to their respective variables. The left side can be integrated using the substitution u = cosx+sinx+y, which will give you du = (-sinx+cosx+1)dx. The right side can be integrated using the substitution v = cosx-sinx, which will give you dv = (-sinx-cosx)dx.

After integrating and substituting back in the original variables, you will get:

ln|cosx+sinx+y| = ln|cosx-sinx| + C

Where C is the constant of integration. This can be rewritten as:

cosx+sinx+y = C(cosx-sinx)

Which is the general solution to the given differential equation in implicit form.
 

FAQ: General solution of a differential equation

1. What is a general solution of a differential equation?

A general solution of a differential equation is a function that satisfies the differential equation for all possible values of the independent variable. It contains a constant of integration that allows for infinitely many solutions to the equation.

2. How do you find the general solution of a differential equation?

To find the general solution of a differential equation, the first step is to separate the variables and integrate both sides. This will result in an equation with a constant of integration. Then, any initial conditions can be used to solve for the value of the constant, resulting in the general solution.

3. Can a general solution of a differential equation have multiple constants of integration?

Yes, a general solution of a differential equation can have multiple constants of integration. This is because there may be more than one way to solve the differential equation, leading to different constants of integration in the final solution.

4. How is a particular solution different from a general solution of a differential equation?

A particular solution of a differential equation is obtained by using specific initial conditions to solve for the constant of integration in the general solution. It is a specific solution that satisfies both the differential equation and the given initial conditions.

5. Can a general solution of a differential equation be expressed in terms of elementary functions?

In some cases, yes, a general solution of a differential equation can be expressed in terms of elementary functions such as polynomials, exponential functions, and trigonometric functions. However, there are many differential equations that do not have closed-form solutions and require more advanced techniques to find the general solution.

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