The discussion revolves around the integration of a first-order linear differential equation of the form dy/dx = 2y + 4x + 10, focusing on the use of an integrating factor to solve the equation.
Some participants have provided guidance on the integration process and the use of integration by parts. There is an ongoing exploration of how to handle the integral of the right side, with no explicit consensus reached on the next steps.
Participants express uncertainty about the integration of the right-hand side and the specifics of applying integration techniques, indicating a need for further clarification on these points.
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?