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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