# Find (if it exists) the solution for the differential equation.

• Lengalicious
In summary, When faced with a differential equation with both x and y in the function, it is important to find the domain using the theorem of existence and unicity. After determining the existence of a unique solution, separation of variables can be used to find the solution. To ensure the validity of the solution, it should be substituted back into the original equation. To solve for y, the substitution A = sin(y) can be used, and the initial values can be used to find the value of C.
Lengalicious
Find (if it exists) the solution for the differential equation:
dy/dx = -2xtan(y)
given the initial value y(0) = /4

My steps in tackling this, by using the theorem of existence and unicity I would take domain of f(x,y) and the domain of the partial derivative, then figure out the common interval and see whether the initial value is within this interval to find whether a unique solution exists. However, my dilemma. I understand how to find the domain for say dy/dx = √x because its just x in the function, so y would be all real values and x would be ≥ 0. But in my above example there is x AND y in the function, how do I find the domain?

Once I figure out whether it has unique solution or not I think I can find solution.

Hi Lengalicious!
Lengalicious said:
Once I figure out whether it has unique solution or not I think I can find solution.

uhh?

find the solution first … separate the variables!

But how can I find the solution if I don't know if one exists? :s

Once you do separation of variables, if you're still not convinced that you know there's a solution, just take the solution you got from separation of variables and plug it into the differential equation to make sure it really works

Lengalicious said:
But how can I find the solution if I don't know if one exists? :s

how can you find the silver lining if you don't know if one exists?

ok so after i separate variables and integrate i get ln(sin(y)) = -x^2 + C , how do I solve for y? Also was my integration correct?

yes it is correct.

Try by setting sin(y) = A and solve for A. Then substitute back and solve for y.

Ok and once I've done that do I insert the intial values for the particular solution and that is the final answer?

Yes you insert your initial values and find the C. Then your done

Ok thank you very much

Just to double check when substituting A you get ln(A)=-x2+C

I still not sure how to solve for x because the C is there, without the c it is just A = e-x2, but with the C there what do I do, or do i ignore the C ?

Well just do the same with the C there, what difference does it make?

So ln(A)=-x^2+c
A=e^(c-x^2)
sin(y)=e^(c-x^2)
y=arcsin*e^(c-x^2)

Last edited:
I assume you mean y=arcsin(e^(c-x^2)) ?
Now to find the C, plug in the values in this equation
ln(sin(y)) = -x^2 + C
as it is way easier to solve for C

## 1. What is a differential equation?

A differential equation is an equation that describes the relationship between a function and its derivatives. It is used to model various physical, chemical, and biological phenomena in the natural world.

## 2. How do you find the solution to a differential equation?

The process of finding a solution to a differential equation depends on the type of equation. In general, it involves manipulating the equation using mathematical techniques to isolate the dependent variable and solve for its value.

## 3. Are there different methods for solving differential equations?

Yes, there are various methods for solving differential equations, including separation of variables, substitution, integration, and using specific formulas for certain types of equations. The method used depends on the type and complexity of the equation.

## 4. Can differential equations have multiple solutions?

Yes, differential equations can have multiple solutions. In fact, most differential equations have an infinite number of solutions, which can be found by including arbitrary constants in the solution.

## 5. How do you know if a solution to a differential equation is correct?

To check if a solution to a differential equation is correct, you can substitute the solution into the original equation and see if it satisfies the equation. This process is called verifying the solution.

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