Second order linear system and power series: Differential Equations

In summary, we are trying to find a third degree polynomial approximation for the general solution to a differential equation with a logarithmic term. The corresponding homogeneous equation has a solution of y(t) = k1e-t +k2e-2t, and we can use a polynomial of the form at^3/3 - bt^2/2 + ct to approximate the solution to the original equation. Plugging in values for a, b, and c gives us a good approximation, but there will still be higher order terms remaining. The third degree polynomial is an approximation, not the exact solution to the equation.
  • #1
clarineterr
14
0

Homework Statement


Find a third degree polynomial approximation for the general solution to the differential equation:

[tex]\frac{d^{2}y}{dt^{2}}[/tex] +3[tex]\frac{dy}{dt}[/tex]+2y= ln(t+1)

Homework Equations


Power series expansion for ln(t+1)


The Attempt at a Solution



The system to the corresponding homogeneous equation [tex]\frac{d^{2}y}{dt^{2}}[/tex] +3[tex]\frac{dy}{dt}[/tex]+2y = 0

is y(t) = k1e-t +k2e-2t

Then I guessed[tex]\frac{ at^{3}}{3}[/tex]-[tex]\frac{bt^{2}}{2}[/tex]+ct as a solution for the original equation. Plugging this in I got a=1/2, b=2,c=2/3

But then I still have the t[tex]^{4}[/tex], t[tex]^{5}[/tex] terms, etc left in the equation. I am not quite sure how a third degree polynomial can be a solution to this equation.
 
Physics news on Phys.org
  • #2
clarineterr said:
I am not quite sure how a third degree polynomial can be a solution to this equation.
Because it will be an approximation not really the solution itself.
 

FAQ: Second order linear system and power series: Differential Equations

What is a second order linear system?

A second order linear system is a type of differential equation that involves the second derivative of an unknown function. It is written in the form of y'' + p(t)y' + q(t)y = g(t), where p(t) and q(t) are functions of t and g(t) is a known function. This type of system can be used to model various physical, biological, and economic phenomena.

How do you solve a second order linear system?

To solve a second order linear system, you need to find the general solution of the equation. This can be done by using the method of undetermined coefficients or the method of variation of parameters. Both methods involve finding the complementary solution, which satisfies the homogeneous equation (y'' + p(t)y' + q(t)y = 0), and the particular solution, which satisfies the non-homogeneous equation (y'' + p(t)y' + q(t)y = g(t)). The general solution is then the sum of the complementary and particular solutions.

What are power series solutions for second order linear systems?

Power series solutions are a type of solution for second order linear systems that involve expressing the unknown function as a power series. This means that the function is written as a sum of terms with increasing powers of the independent variable. This method is useful for solving differential equations that cannot be solved using other methods, such as when the coefficients p(t) and q(t) are not constant.

How do you find a power series solution for a second order linear system?

To find a power series solution for a second order linear system, you first need to rewrite the equation in terms of the independent variable and its derivatives. Then, you can assume that the solution can be expressed as a power series and substitute this into the equation. By equating coefficients of the same power, you can find a recurrence relation that allows you to determine the coefficients of the power series. The series will converge if certain conditions are met, such as that the coefficients are bounded or decreasing.

What are some applications of second order linear systems and power series?

Second order linear systems and power series have many applications in various fields. In physics, they can be used to model oscillatory motion, electric circuits, and quantum mechanics. In biology, they can be used to model population growth and predator-prey relationships. In economics, they can be used to model economic growth and inflation. They are also used in engineering for control systems and signal processing. Additionally, power series solutions have applications in computing and data analysis, such as in numerical methods and data fitting.

Back
Top