9.1 Solving Differential Equations Mon Jan 04 Do Now Find the original function if F’(x) = 3x + 1 and f(0) = 2

Quiz Review (If everyone missing it taking it now)

Differential Equations A differential equation is an equation that involves an unknown function y = y(x) and one of its derivatives. A solution is a function y = f(x) that satisfies the equation.

Properties of Differential Equations The order of a differential equation is the order of the highest derivative appearing in the equation A differential equation is linear if all derivatives in the equation are considered linear – The independent variable (x) does not have to be linear

Exs Diff eqOrderLinear or no? 1 st Linear 1 st Nonlinear 2 nd Linear 3 rd Nonlinear

Separable Differential Equations A differential equation is separable if we can separate the variables into the form – All y variables on one side – All x variables on the other side – Move by multiplication or division only

Separable Differential Eqs To separate x and y, they must be separated by multiplication or division You need to factor either x or y then use multiplication or division to separate them

Ex 5.1 Separable Differential Equation Determine if is separable

Ex 5.2 Not separable Determine ifis separable

The Initial Value Problem (IVP) 1) Separate the variables – Factor – Multiply and divide 2) Integrate both sides with respect to each variable 3) Solve for y (if possible) 4) Plug in for x and y, and solve for C 5) Plug the value for C into step 3

Ex 5.4 Solve the initial value problem

Ex 5.5 Solve the initial value problem

Ex Solve the initial value problem y’ = -ty, y(0) = 3

Closure Solve the initial value problem HW: p.508 #