Differential Equation and Applications — Chapter Summary

An equation involving dependent variable(s)

An equation involving dependent variable(s), independent variable, and derivatives of dependent variables with respect to the independent variable. Ex

Order is the highest derivative present

Order is the highest derivative present (dy/dx has order 1, d²y/dx² has order 2). Degree is the power of the highest order derivative when freed from

Used when equation can be written

Used when equation can be written as f(x)dx + g(y)dy = 0. Separate variables and integrate both sides. Example: dy/dx = xy becomes dy/y = x dx, then ∫

Equations where f(x

Equations where f(x,y) and g(x,y) are homogeneous functions of the same degree. Solved using substitution y = tx, converting to variable separable for

Form

Form: dy/dx + Py = Q where P, Q are functions of x only. Solution: y(I.F.) = ∫Q(I.F.)dx + c, where I.F. = e^∫P dx. Example: dy/dx + y = x has I.F. = e