How can substitution be used to solve first-order equations?

  • Thread starter strawburry
  • Start date
In summary, the substitution x = at + by + c changes the equation y' = f(at + by + c) to the separable equation x' = a + bf(x). By using this method, the general solution of the equation y' = (y+t)^2 can be found by substituting x=y+t and solving for x'. Finally, by integrating both sides of dx/(1+x^2)=dt, the solution x = cubed root (-3t) can be found.
  • #1
strawburry
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0

Homework Statement



Consider the equation y' = f(at + by + c)
where a, b, and c are constants. Show that the substitution x = at + by + c
changes the equation to the separable equation x' = a + bf(x).
Use this method to find the general solution of the equation y' = (y+t)^2

Homework Equations



n/a

The Attempt at a Solution



not sure where to begin :/
 
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  • #2
strawburry said:

The Attempt at a Solution



not sure where to begin :/


so x = at + by + c

x'=?
 
  • #3
mm so plug in the first equation into where by is?
 
  • #4
Differentiate x with respect to t and solve for y' in terms of x'. Then put that expression for y' into your original equation. Now everything is in terms of x.
 
  • #5
thanks! i got how to do the first part, showing the substitution to show separable equation..

How do i continue on to second part of the question?? SOrryy I am kinda slow :(
 
  • #6
Slow is ok, but your aren't helping yet. As rock.freak667 already asked, what is x'?
 
  • #7
x' = a + by' ??
 
  • #8
strawburry said:
x' = a + by' ??

Sure. Now solve that for y' and put it back into your original equation to eliminate y.
 
  • #9
y' = ( x' - a ) / b

then f(x) = (x'-a)/b

then x' = a + bf(x)
 
  • #10
Well, that's that then, right?
 
  • #11
still not understanding
Use this method to find the general solution of the equation y' = (y+t)^2
 
  • #12
strawburry said:
still not understanding
Use this method to find the general solution of the equation y' = (y+t)^2

Try it. Substitute x=y+t. Do the same thing you just did to get an equation for x. What is it?
 
  • #13
x = y + t
x' = y' +1

y= x^2...x'-1= x^2
 
  • #14
strawburry said:
x = y + t
x' = y' +1

y= x^2...


x'-1= x^2

Nice. Ok, so dx/dt=(1+x^2). That's a separable ode.
 
  • #15
integral ( -x^2 dx) = integral (1)

- 1/3 (x^3) = t

x = cubed root (-3t) ?
 
  • #16
strawburry said:
integral ( -x^2 dx) = integral (1)

- 1/3 (x^3) = t

x = cubed root (-3t) ?

Oh, come on, that's just silly. dx/(1+x^2)=dt. Integrate both sides. I was sort of hoping you knew separable ODE's. Heard of them?
 
  • #17
haha just started learning them :P
 
  • #18
strawburry said:
haha just started learning them :P

Now's the time to use them.
 
  • #19
tytyty
 

FAQ: How can substitution be used to solve first-order equations?

What is a first-order equation?

A first-order equation is an equation that involves only one independent variable and its first derivative. It can be written in the form of y' = f(x,y), where y' is the first derivative of y with respect to x.

How do you solve a first-order equation?

The general method for solving a first-order equation is to separate the variables and integrate both sides with respect to the independent variable. This involves isolating the variables on opposite sides of the equation and then integrating both sides. However, there are also specific techniques and formulas that can be used for certain types of first-order equations.

What are some real-life applications of first-order equations?

First-order equations are used in many fields of science, such as physics, chemistry, biology, and economics. Some examples of real-life applications include modeling population growth, predicting the decay of radioactive substances, and analyzing the motion of objects under the influence of forces.

What is the difference between linear and nonlinear first-order equations?

A linear first-order equation is one in which the dependent variable and its first derivative appear only in a linear form, while a nonlinear first-order equation involves nonlinear terms. This means that a linear equation can be solved using basic integration techniques, while a nonlinear equation may require more advanced methods.

Can first-order equations have multiple solutions?

Yes, a first-order equation can have multiple solutions. This is because the process of solving a first-order equation involves finding the general solution, which is a family of solutions that includes all possible solutions. Different initial conditions or boundary conditions can lead to different specific solutions within this general solution.

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