$\textsf{multiply thru with $x^2$}$karush said:$\textsf{find the general solution}$
$$xy'+2y=e^x$$
$\textsf{divide thru by x}$
$$y' +\frac{2}{x}y=\frac{e^x}{x}$$
$\textsf{Find u(x)}$
$$\displaystyle u(x)=\exp\int\frac{2}{x} \, dx=e^{\ln x^2}=x^2$$
The general solution to this differential equation is y(x) = Cx^2 + e^x/2, where C is an arbitrary constant.
To find the general solution, we first solve for y' by dividing both sides of the equation by x, giving us y' + 2y/x = e^x/x. Next, we use the integrating factor method to solve for y. Once we have the general solution, we can add any arbitrary constant to it to account for all possible solutions.
The integrating factor method is a technique used to solve first-order linear differential equations. It involves multiplying both sides of the equation by an integrating factor, which is calculated using the coefficient of the dependent variable. This reduces the equation to a form that can be easily integrated.
Sure. Using the equation xy'+2y=e^x, the integrating factor is calculated as e^(2lnx) = x^2. Multiplying both sides of the equation by x^2, we get x^2y'+2x^2y=e^xx^2. The left side of the equation can be written as (x^2y)' = e^xx^2. Integrating both sides and adding an arbitrary constant, we get the general solution y(x) = Cx^2 + e^x/2.
The arbitrary constant accounts for all possible solutions to the differential equation. Without it, we would only have a particular solution. By adding the arbitrary constant, we are able to find the entire family of solutions to the differential equation.