Solutions of a linear, second-order, homogeneous differential equation

Then, use the fact that the Wronskian of linearly independent solutions is non-zero to show that it is impossible for the Wronskian to disappear in a 3x3 matrix.In summary, the Wronskian of linearly independent solutions is non-zero, which can be shown by expressing a system of equations in matrix form and using the fact that the Wronskian of linearly independent solutions is non-zero. Therefore, a linear, second-order, homogeneous differential equation of the form y''+P(x)y'+Q(x)y=0 cannot have 3 independent solutions.
  • #1
Holly1990
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Homework Statement
Show, by means of the Wronskian, that a linear, second-order, homogeneous diffferential equation of the form y''+P(x)y'+Q(x)y=0 cannot have 3 independent solutions.

The attempt at a solution
I tried by constructing a Wronskian of 3 general solutions expecting the wronskian to disappear but it didn't so I'm guessing I've gone about this the wrong way?
 
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  • #2
Let y1, y2, and y3 be solutions to the differential equation. That means

y1'' + P(x) y1' + Q(x) y1 = 0
y2'' + P(x) y2' + Q(x) y2 = 0
y3'' + P(x) y3' + Q(x) y3 = 0

Try expressing that system of equations in matrix form.
 

1. What is a linear, second-order, homogeneous differential equation?

A linear, second-order, homogeneous differential equation is a mathematical equation that involves a function, its derivatives, and its double derivative. It is called linear because the function and its derivatives appear to the first power, and homogeneous because all terms involve the function and its derivatives only.

2. What is the general form of a solution to a linear, second-order, homogeneous differential equation?

The general form of a solution to a linear, second-order, homogeneous differential equation is of the form y(x) = c1y1(x) + c2y2(x), where c1 and c2 are arbitrary constants and y1(x) and y2(x) are linearly independent solutions to the differential equation.

3. How do you solve a linear, second-order, homogeneous differential equation?

To solve a linear, second-order, homogeneous differential equation, you can use the method of undetermined coefficients or the method of variation of parameters. Both methods involve finding the general solution and then using initial conditions to determine the specific solution.

4. What are the initial conditions of a linear, second-order, homogeneous differential equation?

The initial conditions of a linear, second-order, homogeneous differential equation are values given for the function and its derivatives at a specific point, usually denoted as y(x0) = a and y'(x0) = b. These initial conditions are used to determine the specific solution to the differential equation.

5. What are some real-world applications of linear, second-order, homogeneous differential equations?

Linear, second-order, homogeneous differential equations have many applications in physics, engineering, and other scientific fields. Some examples include modeling oscillations in mechanical systems, analyzing electrical circuits, and predicting population growth in biology. They are also used in control theory and signal processing for designing and optimizing systems.

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