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9.1 Solving Differential Equations Fri March 20 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 eq x y'+ e y = 4 2 x(y') = y + x y'' = (sin x)y' y''' = xsin y 2 x Order 1st 1st 2nd 3rd Linear or no? Linear Nonlinear Linear Nonlinear Separable Differential Equations • A differential equation is separable if we can separate the variables into the form g(y) dy = h(x) dx – 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 dy dx is separable = xy - 2xy 2 Ex 5.2 Not separable • Determine if dy dx is separable 2 = xy - 2x y 2 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 2 dy x +7x +3 dx y2 = , y(0) = 3 Ex 5.5 • Solve the initial value problem dy dx = 9x -sin x y cosy +5e 2 , y(0) = p Ex • Solve the initial value problem y’ = -ty, y(0) = 3 Practice • If time, worksheet Closure • Solve the initial value problem dy dx = -3 x 2 y , y(1) = 0 • HW: p.507 #1-39 odds