Differential equations. linear system.

In summary, the problem is to show that the solutions of x'=G(t)x form an n-dimensional subspace of C1(R+,Rn). This can be demonstrated by showing closure, addition, and scalar multiplication, as well as proving that the solutions are linearly independent. This can be done by defining n different solutions, each with a different initial condition, and showing that they satisfy the differential equation and initial condition.
  • #1
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Homework Statement



G(t) is nxn matrix depends on t.
Show that solutions of x'=G(t)x form an n-dim subspace of C1(R+,Rn).


The Attempt at a Solution



So I can show closure, addition of solutions returns some combo inside R^n, and same with scalar multiplication. I need to show dimension..
 
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  • #2
Since Gn is an n by n matrix, x must be a column matrix with n rows. Let x1(t) be the solution with x(0)1= (1, 0, 0, ..., 0)T. Let x2(t) be the solution with x2(0)= (0, 1, 0, ..., 0)T. Let x3(t) be the solution with x3(0)= (0, 0, 1, ..., 0)T. Continue until you have xn(t) defined as the solution with xn(t)= (0, 0, 0, ..., 1)T. Show that they are independent, by showing that the only solution to the differential equation with x(0)= (0, 0, 0, ..., 0)T is the 0 function, and that the solution with x(t)= (a1, a2, a3, ..., an) is equal to a1x1(t)+ a2x2(t)+ a3x3(t)+ ...+ anxn(t) by showing that they both satisfy the differential equation and the same initial condition.
 

What is a differential equation?

A differential equation is an equation that relates a function to its derivatives. It is commonly used to model physical phenomena in fields such as physics, engineering, and economics.

What is a linear system of differential equations?

A linear system of differential equations is a set of differential equations that can be written in the form of a matrix equation, where the variables are functions and their derivatives. The coefficients of the variables and their derivatives are constants.

What is the solution to a linear system of differential equations?

The solution to a linear system of differential equations is a set of functions that satisfies all of the equations in the system. This solution can be found by using techniques such as substitution, elimination, and integration.

What is the importance of linear systems of differential equations?

Linear systems of differential equations are important because they can be used to model and predict the behavior of complex systems in various fields. They also have analytical solutions, making them easier to solve compared to non-linear systems of differential equations.

What are some common applications of linear systems of differential equations?

Linear systems of differential equations are commonly used in physics to model motion and in engineering to design control systems. They are also used in economics to model economic growth and in biology to model population dynamics.

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