Final answer:
The general solution of the differential equation dy/dx = (x+2y)/(2x-y) can be obtained using the method of exact differential equations. By finding an integrating factor, integrating both sides, and simplifying the resulting equation, we can find the expression for y in terms of x and the constants of integration.
Step-by-step explanation:
The differential equation dy/dx = (x+2y)/(2x-y) can be solved using the method of exact differential equations. To do this, we need to find an integrating factor that makes the equation exact. The integrating factor is given by the formula μ = e∫(P(x)dx+Q(y)dy). In this case, P(x) = 1 and Q(y) = 2, so the integrating factor is μ = ex+2y. Multiplying both sides of the equation by μ gives us the exact equation, which can be written as (x+2y)ex+2ydx + (2x-y)ex+2ydy = 0.
Next, we integrate both sides of the equation. Integrating with respect to x, we get ∫(x+2y)ex+2ydx = ∫0 dx, which simplifies to (x+2y)ex+2y - Cex = F(y), where C is the constant of integration and F(y) represents the integration constant with respect to y. Similarly, integrating with respect to y gives us ∫(2x-y)ex+2ydy = ∫0 dy, which simplifies to (2x-y)ex+2y - De2y = G(x), where D is the constant of integration and G(x) represents the integration constant with respect to x.
Setting the two expressions equal to each other, we have (x+2y)ex+2y - Cex = (2x-y)ex+2y - De2y. Simplifying this equation, we get x+2y-Cex = 2x-y-De2y. Rearranging terms, we have x-y+Cex = y-2y-De2y. Combining like terms, we get x-y+Cex = -y-De2y. Finally, solving for y, we have y = (x+Cex)/(De2y - 1).