General Solution of the first order differential equation

In summary, a general solution of a first order differential equation is a solution that includes all possible solutions of the equation and an arbitrary constant. To find the general solution, variables must be separated and both sides must be integrated. The difference between a general solution and a particular solution is that a particular solution is a specific solution derived from the general solution by substituting values for the arbitrary constant. The arbitrary constant is important in the general solution as it allows for an infinite number of specific solutions. A first order differential equation can have multiple general solutions due to the varying values of the arbitrary constant.
  • #1
Yr11Kid
8
0
dy/dt + y =[tex]\infty[/tex][tex]\sum[/tex]n=1Sin(nt)/n^2
 
Last edited:
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  • #2
The equation is of the form
[tex]
\frac{\mathrm{d}y}{\mathrm{d}t} = P(t)y(t) + Q(t)
[/tex]
with
[tex]P(t)\equiv -1[/tex] and [tex]Q(t):=\sum_{n\ge 1}{\frac{\sin(nt)}{n^2}}[/tex]. So, try with the formula
[tex]
y(t) = \exp\left(\int{P(t)\mathrm{d}t}\right)\left(\int{Q(s)\exp\left(-\int{P(s)}\mathrm{d}s\right)\mathrm{d}s})\right) \biggr|_{s=t}
[/tex]
 

Related to General Solution of the first order differential equation

What is a general solution of a first order differential equation?

A general solution of a first order differential equation is a solution that contains all possible solutions of the equation. It includes an arbitrary constant, which allows for an infinite number of specific solutions to the equation.

How do you find the general solution of a first order differential equation?

To find the general solution of a first order differential equation, you must first separate the variables and then integrate both sides. This will give you a general expression that includes the arbitrary constant. You can then use initial conditions to determine the specific solution.

What is the difference between a general solution and a particular solution?

A general solution contains all possible solutions to a differential equation, while a particular solution is a specific solution that satisfies given initial conditions. A particular solution is derived from the general solution by substituting specific values for the arbitrary constant.

Why is it important to include an arbitrary constant in the general solution?

The arbitrary constant in the general solution allows for an infinite number of specific solutions to the differential equation. This allows for a more complete understanding of the behavior of the equation and its solutions.

Can a first order differential equation have more than one general solution?

Yes, a first order differential equation can have multiple general solutions. This is because the arbitrary constant can take on different values, resulting in different general expressions that contain all possible solutions to the equation.

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