Use the Laplace transform to solve the given system of differential equations. (d x)/(d t)=-

Use the Laplace transform to solve the given system of...

Key Concepts

Inverse Laplace Transform

Once the Laplace transforms of the unknown functions have been determined by solving the algebraic equations, the inverse Laplace transform is used to convert the solution back into the time domain. This step often involves techniques such as partial fraction decomposition and the use of Laplace transform tables to match known inverse transforms, ultimately yielding the solution to the original differential equations.

Initial Value Problems (IVPs)

Initial value problems are differential equations that include conditions specifying the state of the system at the beginning of the observation (usually at time t = 0). The Laplace transform method is particularly effective in dealing with IVPs because the initial conditions appear naturally in the transformation, thus bypassing the need to separately determine constants of integration after solving the transformed equations.

Laplace Transform

The Laplace transform is an integral transform used to convert functions from the time domain into the complex frequency domain, simplifying the process of solving differential equations. By transforming derivatives into algebraic expressions, it allows for the treatment of differential equations as algebraic equations, which are typically easier to manipulate and solve.

Operational Properties of Laplace Transforms

One of the essential aspects of the Laplace transform is its ability to transform differential equations using properties such as the derivative rule (i.e., L{f'} = sF(s) - f(0)), which incorporates initial values directly into the transform. These operational properties extend to higher order derivatives and make the method particularly powerful in dealing with initial value problems.

Systems of Differential Equations

Systems of differential equations involve multiple interdependent equations that describe the evolution of several variables simultaneously. By applying the Laplace transform to each equation in the system, the system is converted into a set of linear algebraic equations in the Laplace domain. This conversion facilitates the use of linear algebra techniques to solve for the unknown transforms.

Transcript

00:01 Hi everyone, so today we are going to solve these given differential equation, the system of given differential equation, that is dx divided by dt is equal to negative x plus y, divide and the other is d by d by dt that is equal to 2x.

00:26 And you have been given the condition that x of 0 is equal to 0 and y of 0 is equal to positive 1.

00:45 Now what we have to do over here we have to use the laplace transform in order to solve this system of given differential equation.

00:59 Now how will we solve it? so solution for this question is very simple.

01:12 Now first of all, we have to take the laplace transform of both sides of the both equations.

01:21 So let's take it.

01:24 Now, what we are going to do over here, we are going to use the linearity of the laplace.

01:37 Transform and we would have x prime is equal to negative x plus y so this is the given equation right and now we are going to take the lap plus transform so over here we have this that is equal to negative l of x sorry square bracket plus l so this is the first equation.

02:21 Next we have the second equation that is y prime is equal to 2x.

02:28 So this will become taking the lap plus transform negative y or 0 is equal to 2 l x.

02:42 Fine.

02:44 And what are we going to do next we are going to take group coefficients.

02:48 So we'll take the group coefficients, we would get s plus 1, l of x, that is negative, ly is equal to 0...

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