2nd order inhomo differential eq.

In summary, the conversation discusses finding a particular solution for the nonhomogeneous differential equation y'' = t^2. The method of undetermined coefficients is suggested, where a particular solution of At^4 is tried. The general solution is then found to be y = c1 + c2t + At^4. It is also noted that this method is not a mere guess, but a reliable method for solving constant-coefficient-linear differential equations.
  • #1
projection
48
0

Homework Statement



[tex]y'' = t^2[/tex]


The Attempt at a Solution



The general solution to the homogenous diff equation is [tex]y(t)= C1 + tC2[/tex] i beliee. The particular solution is where i am having trouble.

the guess is of the form [tex]\alpha t^2 + \beta t + \gamma[/tex] ... but taking 2 deriatives leads to just [tex]2\alpha = t^2[/tex] ... this can only be zero if [tex]\alpha = \frac{t^2}{2}[/tex] but this is something i haven't encountered as the constants are usually numeric values... and what about the rest of constants of beta and gamma?
 
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  • #2
Try a particular solution of yp = At4.

The general solution will be the complementary solution (c1 + c2t) plus the particular solution.
 
  • #3
Mark44 said:
Try a particular solution of yp = At4.

The general solution will be the complementary solution (c1 + c2t) plus the particular solution.

i am not following... i am trying to find a particular solution... am i not supposed to find a polynomial of similar degree with unknown coefficients and finding derivates and then equating it to the regular diff equation to solve for the coefficients... trying a particular solution, i am unfamiliar with this method.
 
  • #4
You need to look at nonhomogenous equations in two separate parts: the homogeneous equation and then the nonhomogeneous equation.

For your problem, the homogeneous problem is y'' = 0, to which the solutions are y = c1 + c2t.

Any linear combination of the functions 1 and t (i.e., any sum of constant multiples of 1 and t) will end up at 0 when you take the derivative twice, so a particular solution can't be 1, or t or any constant multiple of these.

One function whose 2nd derivative is t2 is t4, so for a particular solution to the nonhomogeneous problem, I suggest trying yp = At4. Plug this into the nonhomog. equation and solve for the value of A that works (this is called the method of undetermined coefficients, IIRC.

Your general solution to the nonhomog. equation will be y = c1 + c2t + At4, and your job is to figure out what A needs to be.

There's a lot more I could say, but it's probably in your textbook and you haven't come to it yet.
 
  • #5
@projection don't believe when they say you are "guessing" an answer - it's not a guess it's a METHOD. If you build a template by the correct method it will ALWAYS give you a correct answer in constant-coefficient-linear DEs (that the method can handle) :)

You didn't build the correct template because you did it not by the METHOD.
I hope Mark44 explained it in a way you will understand :)
 

1. What is a 2nd order inhomogeneous differential equation?

A 2nd order inhomogeneous differential equation is a mathematical equation that involves a second derivative of a function and also includes a non-zero function on the right-hand side. It is typically written in the form y'' + p(x)y' + q(x)y = g(x), where p(x) and q(x) are functions and g(x) is a non-zero function.

2. How is a 2nd order inhomogeneous differential equation different from a 2nd order homogeneous differential equation?

A 2nd order homogeneous differential equation has a zero function on the right-hand side, while a 2nd order inhomogeneous differential equation has a non-zero function. This means that the solutions to a homogeneous equation only involve the functions p(x) and q(x), while the solutions to an inhomogeneous equation also involve the function g(x).

3. What are some real-life applications of 2nd order inhomogeneous differential equations?

2nd order inhomogeneous differential equations are commonly used in physics, engineering, and other sciences to model systems that involve acceleration or oscillation. For example, the motion of a spring-mass system can be described by a 2nd order inhomogeneous differential equation.

4. How do you solve a 2nd order inhomogeneous differential equation?

There are several methods for solving 2nd order inhomogeneous differential equations, including the method of undetermined coefficients, variation of parameters, and Laplace transforms. These methods involve finding a particular solution and a complementary solution, and then combining them to form the general solution.

5. What is the importance of boundary conditions in solving 2nd order inhomogeneous differential equations?

Boundary conditions are important in solving 2nd order inhomogeneous differential equations because they help determine the specific solution to the equation. Without boundary conditions, there may be an infinite number of solutions to the equation. By specifying initial conditions or boundary values, the solution can be narrowed down to a specific function that satisfies the given conditions.

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