Except when the exercise indicates otherwise, find a set of solutions. y(y^3-x) d x+x(y^3+x) d y

step 1 Hello, hope you're doing well. So we've got our two equations here that form our system of equat

Except when the exercise indicates otherwise, find a set of...

Key Concepts

Ordinary Differential Equations (ODEs)

An ordinary differential equation (ODE) is an equation that relates some function of one independent variable to its derivatives. Solving an ODE involves finding a function that satisfies the relationship specified by the equation, which often represents a physical, biological or other dynamical system. The process of solving ODEs is fundamental in many areas of science and engineering because it provides insights into system behavior over time.

Exact Differential Equations

An exact differential equation is a first order ODE that can be written in the form M(x, y) dx + N(x, y) dy = 0, where there exists a function F(x, y) such that dF = F_x dx + F_y dy equals the left-hand side of the equation. Verifying exactness involves checking that the mixed partial derivatives of F are equal (i.e., M_y = N_x). When an ODE is exact, its solution can be found by integrating M with respect to x and N with respect to y to obtain an implicit solution F(x, y) = C.

Integrating Factors

An integrating factor is a function, often dependent solely on one variable, that is multiplied by a non-exact differential equation to transform it into an exact one. The need for an integrating factor arises when the original differential equation does not satisfy the condition M_y = N_x. Once the appropriate integrating factor is found and applied, the resulting equation is exact and can be solved by finding a function F(x, y) such that dF = 0, leading to an implicit general solution.

Exactness Condition

The condition for an ODE of the form M(x, y) dx + N(x, y) dy = 0 to be exact is that the partial derivative of M with respect to y must equal the partial derivative of N with respect to x (i.e., M_y = N_x). This criterion is essential for determining whether the method of exact equations can be applied directly or if an integrating factor is necessary.

Implicit Solutions

In many cases of solving differential equations, especially when dealing with exact equations, the solution is expressed implicitly in the form F(x, y) = C, where C is an arbitrary constant. Implicit solutions represent the relationship between the variables x and y without necessarily solving for y explicitly. This approach is common because it often leads to simpler or more natural expressions of the general solution.

Transcript

00:01 Hello, hope you're doing well.

00:02 So we've got our two equations here that form our system of equations.

00:05 So we're going to solve the system of equations using the addition method.

00:09 So we add these two equations together.

00:12 Everything's already lined up for us, which is nice.

00:14 So we've got a minus y plus y here, just cancel out and go to zero.

00:19 So we're left with x cubed plus x squared is equal to 0 plus 0, which is just 0.

00:25 So we can factor out an x squared here.

00:28 So we've got x squared and x plus 1.

00:30 Is equal to 0.

00:33 So now in order to solve for x, we can set each of these factors equal to 0 and then solve for x.

00:39 To see what x value will make this equation true.

00:42 So our first equation is going to be got our first factor x squared, and we set that equal to 0.

00:48 And we've got our second factor, x plus 1, and we set that equal 0.

00:52 So for this first equation, we're going to take the square to both sides, get x is equal to 0.

00:57 So this is our x1.

00:59 This is our first x value that's part of our solution to this system of equations.

01:04 So now, moving on to our second equation, we can subtract one from both sides, get x is equal to minus 1.

01:11 So this is x2.

01:12 So these are our two x values that are part of the x components of our set of solutions to the system of equations.

01:20 So now what we want to do is we want to figure out what y values correspond to each of these x values...

Please give Ace some feedback