General Solution of the first order differential equation

In summary, the conversation discusses using the equation y(t) = yh(t) + yp(t) to find a particular solution to dy/dt + y = sin(nt)/n^2. The suggested method is to find particular solutions for each value of n and then sum them together to get the general solution. It is also mentioned that this is an integrating factor question and the general solution to the equation y'+y=0 is y(t)=Ce^{-t}.
  • #1
Yr11Kid
8
0
dy/dt + y = [tex]\infty[/tex] [tex]\sumSin(nt)/n^2[/tex] n=1


Ok still a bit new with all these symbols and stuff but that is the basic jist of it.

y(t) = yh(t) + yp(t) it what i thought about using to start off with, yh(t) = Acos2t + Bsin2t.
Then subbing yp(t) into the differential equation. Not really sure about how to do this, help much appreciated :)
 
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  • #2
Hey everyone,
This question kind of confused me, i think the general idea is to use y(t) = yh(t) + yp(t) and basically saying yh(t) = Acos2t + Bsin2t

to find yp(t) subbing into the differential equation and equating.

help much appreciated :)
 

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  • #3
Here's an idea. It shouldn't be too hard to find particular solutions to dy/dt + y = sin(nt)/n^2, where n is an unspecified constant. Do that and then sum the solutions. You'll end up with a weird infinite series, but it should be convergent because of the n^2 in the denominator. And this is probably the best you can do, since the problem itself has a series as the nonhomogeneous term.
 
  • #4
Do you mean:
[tex]
\frac{dy}{dt}+y=\sum_{n=1}^{\infty}\frac{\sin (nt)}{n^{2}}
[/tex]
This is an integrating factor question.
 
  • #5
The general solution to the equation y'+ y= 0 is [itex]y(t)= Ce^{-t}[/itex]

To find the general solution to the entire equation, look for a solution of the form
[tex]\sum_{n=1}^\infty A_n sin(nt)+ B_n cos(nt)[/tex]

That will give you as sequence of equations to solve for [itex]A_n[/itex] and [itex]B_n[/itex]. Once you have that, add to [itex]Ce^{-t}[/itex].
 
  • #6
I am combining the two threads on the same thing.
 

FAQ: General Solution of the first order differential equation

What is a first order differential equation?

A first order differential equation is an equation that relates a function to its derivative. It can be written in the form dy/dx = f(x), where y is the dependent variable and x is the independent variable.

What is a general solution?

A general solution is a solution to a differential equation that contains all possible solutions. It includes a constant of integration, so it can be used to find specific solutions by plugging in different values for the constant.

How do you solve a first order differential equation?

To solve a first order differential equation, you can use a variety of methods such as separation of variables, integrating factors, or substitution. The specific method used depends on the form of the equation.

What is the role of initial conditions in finding a particular solution?

The initial conditions, also known as boundary conditions, provide information about the specific solution to a differential equation. They are typically given in the form of values for the dependent variable and/or its derivative at a specific point, and are used to determine the constant of integration in the general solution.

How are first order differential equations used in science?

First order differential equations are used in many areas of science, such as physics, chemistry, biology, and engineering. They can be used to model a wide range of real-world phenomena, from population growth to chemical reactions to electric circuits. They also play a crucial role in developing mathematical models and predicting future behavior in various scientific fields.

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