Write the equation as Mdx + Ndy = 0, and you'll see that this equation is exact, since My = Nx.illidari said:Homework Statement
x(dy/dx) = 2xe^x - y + 6x^2
Homework Equations
The Attempt at a Solution
I am trying to show this is an exact differential (book has an answer, it must be )
I rearranged the problem to be :
2xe^x - y + 6x^2 dx = x dy
Partial derivatives (using & for that symbol that represents partial)
&m/&y = -1
&m/&x = 1
-1 doesn't equal 1
Any idea where my mistake is?
An exact differential equation is a type of differential equation where the total differential of the equation can be expressed as the sum of two functions, one of which is the derivative of the other. In other words, the equation can be rewritten in the form of M(x,y)dx + N(x,y)dy = 0, where M and N are functions of x and y.
To solve an exact differential equation, you must first check if the equation is exact by verifying if the partial derivatives of M and N with respect to x and y are equal. If they are equal, you can then use the method of separation of variables to find a general solution. This involves rewriting the equation in the form of M(x,y)dx + N(x,y)dy = 0 and then integrating both sides with respect to x or y.
The specific solution to this exact differential equation is y = 6x^2 + Cx + 2e^x, where C is a constant. This can be found by using the method of separation of variables and integrating both sides with respect to x.
Solving exact differential equations is important in various fields of science and engineering, as it allows us to model and understand the behavior of complex systems. It is particularly useful in physics, chemistry, and economics, among other disciplines, where many natural processes can be described by differential equations.
Yes, exact differential equations can have multiple solutions. This is because when solving the equation, we usually have to integrate both sides, and the integration constant can take on different values, resulting in different solutions. In some cases, there may also be a family of solutions that can be expressed in terms of arbitrary functions.