Differential Equation For the below ordinary differential eq

Differential Equation

For the below ordinary differential equation with initial conditions, state the order and dtermine if the equation is linear or nonlinear. Then find the solution of the ordinary differential equation, and apply the initial conditions. Verify your solution.

Differential Equation For the below ordinary differential equation with initial conditions, state the order and dtermine if the equation is linear or nonlinear. Then find the solution of the ordinary differential equation, and apply the initial conditions. Verify your solution. frac{dy}{dx} - frac{y}{x} = xe^{x}, y(1) = e-1

Solution

dy/dx - y/x = xe^x

the function with y is -1/x

Multiplicating factor = e^(?-dx/x) = e^ -ln|x| = e^ ln|1/x| = 1/x
So, the equation becomes:
1/xdy/dx - y/x^2 = xe^x (1/x)
1/xdy/dx - y/x^2 = e^x
d/dx (y/x) = e^x
d (y/x) = e^x dx
y/x = e^x + C [Integrating]

y = xe^x + Cx

y(1)=e-1

e-1=e+c

c=-1

So, final solution is y = xe^x -x