Solving First-Order Linear Equation with Reversed Roles

In summary, the conversation discusses a first-order linear differential equation and finding the inverse function. The equation is given with y as a function of x, but can be reversed to make x a function of y. The conversation also mentions using Mathematica and solving the equation using linear and quadratic methods.
  • #1
awelex
44
0
Hi,

I have a differential equation that I just can't seem to solve. Now here's the deal: I'm sure there are advanced methods that would easily solve this equation, but the equation is in the chapter on First-Oder Linear Equations, so it shouldn't be anything fancy. There's even a hint: it says that "the roles of the independent and dependent variables may be reversed." What is that supposed to mean?

I tried getting the solution with Mathematica first, and then working backwards, but to no avail. I don't even know where to start.

Here's the equation:

dy/dx = 1/(e^(4y) + 2x)

Any pointers? Thanks!
 
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  • #2


With first order differential equations, which variable is a function of the other is not really important. That particular equation is given with y a function of x and is very badly non-linear. But if you "flip it over", it becomes
[tex]\frac{dx}{dy}= 2x+ e^{4y}[/tex]
which is a linear equation that is relatively easy to solve for x as a function of y. That's what they mean by "the roles of the independent and dependent variables may be reversed."
 
  • #3


Just put x=v & y=u & solve for du/dv.
 
  • #4


@HallsosIvy: Thanks, that makes sense. I solved the resulting DE and get

x = 1/2 * e^(4y) + C*e^(2y)

But how do I find the inverse function of that?
 
  • #5


Do you have any reason to want to? That is a perfectly good implicit definition of y.
 
  • #6


x = 1/2 * e^(4y) + C*e^(2y)
But how do I find the inverse function of that?
Let t=e^(2y)
x = (1/2)*t² +C*t
Solve for t
Then y = (1/2)*ln(t)
In fact, ln(abs(t))
 
  • #7


Your latter "in fact" comment is not strictly necessary, JJ, since t is, by def.>0
 
  • #8


@Awelex
Which method did you use to solve this equation?
 
  • #9


sgtkt said:
@Awelex
Which method did you use to solve this equation?
x = (1/2)*t² +C*t
(1/2)t²+Ct-x =0
I suppose that you know how to solve a*X²+b*X+c=0 for X
a=1/2 ; b=C : c=-x and t=X
 
  • #10


I mean which method, linear or separable
 
  • #11


sgtkt said:
I mean which method, linear or separable

Linear method to solve dx/dy -2x = exp(4y) because it is a linear ODE
Then quadratic resoltion to inverse the x(y) function.
 
  • #12


Thankyou
 

What is a first-order linear equation with reversed roles?

A first-order linear equation with reversed roles is an equation where the dependent and independent variables are switched. This means that the variable being solved for is on the right side of the equation, while the variable that is known is on the left side.

What is the process for solving a first-order linear equation with reversed roles?

The process for solving a first-order linear equation with reversed roles involves isolating the variable being solved for on one side of the equation and all other terms on the other side. Then, the variable is solved using algebraic operations such as addition, subtraction, multiplication, and division.

Are there any special rules or techniques for solving first-order linear equations with reversed roles?

Yes, there are a few special rules and techniques that can be used to solve first-order linear equations with reversed roles. These include the inverse property of addition and multiplication, the distributive property, and the commutative property of addition and multiplication.

What are some common mistakes to avoid when solving first-order linear equations with reversed roles?

Some common mistakes to avoid when solving first-order linear equations with reversed roles include forgetting to switch the roles of the variables, not using the correct order of operations, and making errors in basic algebraic operations.

How can solving first-order linear equations with reversed roles be applied in real-life situations?

Solving first-order linear equations with reversed roles can be applied in various real-life situations, such as calculating interest rates, solving for unknown quantities in physics equations, and determining the relationship between two variables in a scientific experiment.

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