A question on second order linear equations

In summary, the technique for solving linear ODEs with constant coefficients is to find the complementary solution.
  • #1
mech-eng
828
13
Hi, all. While solving a second order linear differential equation why do we have to use linear independent but two solutions. For example, when solving y''- y = 0 , y(0) = 5 and y'(0) = 3 , we use ex and e-x and then we write y = c1*ex+ c2*e-x-
 
Physics news on Phys.org
  • #2
That's because any linear combination of a solution of the DE is a solution of the DE.
Initial conditions restrict the set of possible solutions.
 
  • #3
That's because any linear combination of a solution of the DE is a solution of the DE.
Initial conditions restrict the set of possible solutions.
 
  • #4
maajdl said:
That's because any linear combination of a solution of the DE is a solution of the DE.
Initial conditions restrict the set of possible solutions.

I am especially interested in the number "two", "two solutions". Why we derive general solution
from any two?
 
  • #5
You seem to be confused about the solutions to ODEs.

For the equation y" - y = 0, with y(0) = 5 and y'(0) = 3, we have a homogeneous ODE for which we find the complementary solution by assuming y = e^(rx). Differentiating and substituting into the original ODE gives:

r^2 * e^(rx) - e^rx = 0

Dividing both sides by e^rx (which is always > 0, for any x), we obtain the characteristic polynomial

r^2 - 1 = 0 or r^2 = 1, which has solutions r = 1 and r = -1.

This means that e^x and e^-x are both solutions to the original ODE. Since the original ODE is linear, the sum of the individual solutions obtained from solving the characteristic polynomial must also be a solution, thus

y = C1*e^-x + C2 * e^x

By applying the initial conditions y(0) = 5 and y'(0) = 3, the coefficients of the general solution C1 and C2 can be determined.
 
  • #6
How do we guess that complementary solution? In my example you did it as erx
 
  • #7
mech-eng said:
How do we guess that complementary solution? In my example you did it as erx

That's the standard technique for solving linear ODEs with constant coefficients. See:

http://en.wikipedia.org/wiki/Linear_differential_equation

Section: Homogeneous equations with constant coefficients
 
  • Like
Likes 1 person

Related to A question on second order linear equations

What is a second order linear equation?

A second order linear equation is a type of mathematical equation that involves two derivatives of a variable, such as y'' and y', and can be written in the form ay'' + by' + cy = f(x), where a, b, and c are constants and f(x) is a function of x.

What is the general solution to a second order linear equation?

The general solution to a second order linear equation is a formula that includes two arbitrary constants and can be used to find all possible solutions to the equation. It is usually written as y = C1y1(x) + C2y2(x), where C1 and C2 are the arbitrary constants and y1(x) and y2(x) are linearly independent solutions to the equation.

How do you solve a second order linear equation with constant coefficients?

To solve a second order linear equation with constant coefficients, you can use the method of undetermined coefficients or the method of variation of parameters. These methods involve finding the general solution and then using initial conditions to determine the specific solution that satisfies the given equation.

What is the characteristic equation of a second order linear equation?

The characteristic equation of a second order linear equation is a polynomial equation that is obtained by replacing the derivatives in the equation with the corresponding coefficients. It is usually written as ar^2 + br + c = 0, where r is the variable and a, b, and c are the coefficients of the equation.

What are the different types of solutions to a second order linear equation?

The different types of solutions to a second order linear equation are: real and distinct, real and equal, and complex. Real and distinct solutions occur when the roots of the characteristic equation are two distinct real numbers. Real and equal solutions occur when the roots are the same real number. Complex solutions occur when the roots are complex numbers.

Similar threads

I Number of solutions to the order of the equation
    • Differential Equations
Replies
2
Views
244
I Second order non-homogeneous linear ordinary differential equation
    • Differential Equations
Replies
2
Views
2K
I Fundamental matrix of a second order 2x2 system of ODEs
    • Differential Equations
Replies
2
Views
2K
I Solve second order linear differential equation
    • Differential Equations
Replies
7
Views
1K
I Explanation of all "its linearly independent derivatives"
    • Differential Equations
Replies
3
Views
1K
I Converting Second Order ODE to Hypergeometric Function
    • Differential Equations
Replies
7
Views
2K
I Question about solving linear first order non-homogeneous ODEs
    • Differential Equations
Replies
3
Views
898
I Solving Differential Equation Using Reduction of Order
    • Differential Equations
Replies
2
Views
1K
I Second Order ODE with Exponential Coefficients
    • Differential Equations
Replies
1
Views
2K
MHB Converting a Second-Order IVP into a System of Equations: Can Substitution Help?
    • Differential Equations
Replies
1
Views
966

Hot Threads

Recent Insights

Back
Top