Differential Equations

A differential equation contains derivatives

A differential equation contains derivatives of dependent variable with respect to independent variable. Order is determined by highest derivative (dy

To form differential equation from equation

To form differential equation from equation with n arbitrary constants: differentiate n times and eliminate constants. Example: From y = ax² + b, we g

For equations like dy/dx = f(x)g(y)

For equations like dy/dx = f(x)g(y), separate variables: dy/g(y) = f(x)dx, then integrate both sides. Example: dy/dx = xy becomes dy/y = x dx, integra

Equation dy/dx = f(y/x) is homogeneous

Equation dy/dx = f(y/x) is homogeneous. Solve by substituting y = vx, then dy/dx = v + x(dv/dx). This converts to separable form. Example: dy/dx = (x+

Equation dy/dx + Py = Q

Equation dy/dx + Py = Q (where P,Q are functions of x) is linear. Solution: y·e^∫Pdx = ∫Qe^∫Pdx dx + C. Integrating factor IF = e^∫Pdx. Example: dy/dx