- #1
Romaha_1
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Homework Statement
Solve the differential equation:
Homework Equations
1+(x-x^2*e^(2y))(dy/dx) = 0
The Attempt at a Solution
No idea how to approach this.
It looks pretty clear to me. Basic algebra givesRomaha_1 said:Homework Statement
Solve the differential equation:
Homework Equations
1+(x-x^2*e^(2y))(dy/dx) = 0
The Attempt at a Solution
No idea how to approach this.
HallsofIvy said:It looks pretty clear to me. Basic algebra gives
[tex]x(1- e^{2y})dy/dx= -1[/tex]
and then separate variables as
[tex](1- e^{2y})dy= -\frac{1}{x}dx[/tex]
Now integrate.
tiny-tim said:Hi Romaha_1! Welcome to PF!
(try using the X2 tag just above the Reply box )
Does putting z = e2y help?
JFonseka said:Hmm, there's an x^2 in the equation, I think you missed that bit.
Altabeh said:No it doesn't, does it?
Romaha_1 said:Yes, that's the reason I don't know how to do it. I don't think it is a mistake in the problem, since there is a sign "(very tricky)" after it on the problem set.
JFonseka said:The question isn't necessarily wrong, I'm not too good at this stuff at all. It just seemed to me that HallsofIvy was attempting the separable ODE approach, but missed the x^2.
Romaha_1 said:Yeah, I understand what you meant and agree with you that the separable approach does not work; it was my assumption that the question was wrong.
JFonseka said:I never said the Separable ODE approach doesn't work.
tiny-tim said:Hi Romaha_1! Welcome to PF!
(try using the X2 tag just above the Reply box )
Does putting z = e2y help?
Romaha_1 said:No idea how to approach this.
tiny-tim said:Does putting z = e2y help?
Romaha_1 said:Hi tiny-tim, thank you for the idea!
But I could solve it only when letting z = x2*e2y; I still do not see how this is possible with z = e2y.
A first order differential equation is a mathematical equation that relates a function to its derivative, or rate of change. It is called "first order" because it involves only the first derivative of the function.
The general form of a first order differential equation is dy/dx = f(x,y), where y is the dependent variable, x is the independent variable, and f(x,y) is a function that relates the two variables.
There are several methods for solving first order differential equations, including separation of variables, integrating factors, and using a substitution. The specific method used depends on the form of the equation and the initial conditions given.
First order differential equations are used to model various physical, biological, and social phenomena in science. They provide a mathematical framework for understanding and predicting how systems change over time.
Yes, a first order differential equation can have multiple solutions. This is because the equation represents a family of functions rather than a single function, and different initial conditions can yield different solutions.