How can we solve the differential equation dy/dx= (y/x) +1?

Answer 1

If you rewrite the equation as dy/dx -(1/x)y = 1 using the integrating factor e^-ln(x), you get the solution

y = xln(x) + constant

Answer 2

Hi,

In order to solve this differential equation, first determine whether it is homogeneous. A differential equation is homogeneous if it can be rewritten in the following form:

dy/dx = F(y/x) = F(v), where v = y/x or y = vx

Step 1: Solve y = vx for dy/dx

y = vx

dy/dx = vx' + v'x

dy/dx = v + v'x

Step 2: Compare dy/dx to the differential equation to be solved given v = y/x

v + v' x = y/x + 1

v'x = 1

v' = 1/x

v = ln(x) + C

Step 3: Solve for y given y = vx

y = x ( ln(x) + C )