The left side is actually d/dx(ye-2x), which is different from dy/dx(ye-2x).xlalcciax said:Homework Statement
Integrate dy/dx=2y+4x+10
The Attempt at a Solution
dy/dx-2y=4x+10
Integrating factor = e^(-2)dx=e^-2x
multiply both sides by IF. (e^-2x)dy/dx-2y(e^-2x)=(e^-2x)(4x+10)
dy/dx(e^-2x y)=(e^-2x)(4x+10)
i don't know what to do next.
Mark44 said:The left side is actually d/dx(ye-2x), which is different from dy/dx(ye-2x).
The equation is d/dx(ye-2x) = (4x + 10)e-2x.
Integrate both sides with respect to x, which gives you
[tex]ye^{-2x} = \int (4x + 10)e^{-2x}dx[/tex]
Can you take it from here?
"Integration using integrating factor" is a method used in calculus to solve differential equations. It involves multiplying both sides of the equation by an integrating factor, which helps to simplify the equation and make it easier to integrate.
This method is typically used when solving first-order linear differential equations, where the equation is in the form dy/dx + P(x)y = Q(x). It can also be used for some other types of equations, such as separable or homogeneous equations.
The method works by finding an integrating factor, which is a function that when multiplied by the original equation, makes it easier to integrate. This integrating factor can be found by using an integrating factor formula or by using a method called the "method of inspection". Once the integrating factor is found, the equation can be integrated and solved for the unknown variable.
The main advantage of this method is that it can be used to solve a variety of differential equations, including some that cannot be solved by other methods. It also allows for the use of standard integration techniques, making the process more familiar and easier to follow.
One limitation of this method is that it can only be used for first-order differential equations. It also requires finding an integrating factor, which can sometimes be challenging and time-consuming. Additionally, this method may not always work for more complex or nonlinear equations.