Results 1 to 3 of 3

Thread: First-Order Linear Differential Equation

  1. #1
    Junior Member
    Joined
    Jan 2010
    From
    Florida
    Posts
    29

    First-Order Linear Differential Equation

    I am suppose to solve y'=5y; y(0)=1

    However, I don't understand how to do this as the standard form for this type of equation seems to be: y' + a(t)y = b(t)

    What do I use as b(t)?

    Everything I've done so far works out to zero. Could someone explain?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member Random Variable's Avatar
    Joined
    May 2009
    Posts
    959
    Thanks
    3
    $\displaystyle y'-5y=0 $

    which has the characteristic equation $\displaystyle r-5=0 $

    so the general solution is $\displaystyle C_{1}e^{5t} $

    but $\displaystyle y(0)=1=C_{1} $

    so the final solution is $\displaystyle e^{5t} $
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor harish21's Avatar
    Joined
    Feb 2010
    From
    Dirty South
    Posts
    1,036
    Thanks
    10
    Quote Originally Posted by Quixotic View Post
    I am suppose to solve y'=5y; y(0)=1

    However, I don't understand how to do this as the standard form for this type of equation seems to be: y' + a(t)y = b(t)

    What do I use as b(t)?

    Everything I've done so far works out to zero. Could someone explain?

    $\displaystyle y' = 5y$

    Since this is separable,

    $\displaystyle \int \frac{1}{5y} dy = \int 1 dx$

    $\displaystyle \frac{log(y)}{5} + C_1 = x + C_2$

    $\displaystyle \frac{log(y)}{5} = x + C_3$ Note: $\displaystyle C_3 = {C_2}-{C_1}$ is also a constant

    $\displaystyle log(y) = 5x+ C_4$ note: $\displaystyle C_4 = 5 C_3$ is a constant

    $\displaystyle \therefore y = e^ {5x+{c_4}}$

    $\displaystyle y = e^{5x} \times e^{C_4} $

    Thus,

    $\displaystyle y= C e^{5x}$.......(I) This is what Random Variable has gotten in the above post.

    Note: $\displaystyle C = e^{C_4}$ is also a constant.


    now use y(0)=1 [in(I)] to find C
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. 1st order linear differential equation
    Posted in the Differential Equations Forum
    Replies: 3
    Last Post: Nov 20th 2010, 09:43 AM
  2. First Order Non-Linear Differential Equation
    Posted in the Differential Equations Forum
    Replies: 2
    Last Post: Oct 3rd 2010, 10:46 AM
  3. First Order Linear Differential Equation
    Posted in the Differential Equations Forum
    Replies: 3
    Last Post: Aug 11th 2010, 03:15 AM
  4. Second order non linear differential equation.
    Posted in the Differential Equations Forum
    Replies: 0
    Last Post: Feb 20th 2010, 12:49 PM
  5. Non linear 2nd order differential equation
    Posted in the Differential Equations Forum
    Replies: 8
    Last Post: Jul 27th 2009, 02:37 AM

Search Tags


/mathhelpforum @mathhelpforum