What is a differential equation?
Definition
A differential equation is an equation involving an unknown function y and
its derivatives.
Math 105 (Section 204)
Differential Equations
2011W T2
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What is a differential equation?
Definition
A differential equation is an equation involving an unknown function y and
its derivatives.
Examples: y 00 y + sin xy 0 = 0, y 0 = 6y 2 , y 0 = y (10 − y ).
Math 105 (Section 204)
Differential Equations
2011W T2
1/6
What is a differential equation?
Definition
A differential equation is an equation involving an unknown function y and
its derivatives.
Examples: y 00 y + sin xy 0 = 0, y 0 = 6y 2 , y 0 = y (10 − y ).
Given a differential equation, our goal is to solve it for the unknown
function y . The solution need not be unique, in fact a differential
equation usually has infinitely many solutions!
Math 105 (Section 204)
Differential Equations
2011W T2
1/6
What is a differential equation?
Definition
A differential equation is an equation involving an unknown function y and
its derivatives.
Examples: y 00 y + sin xy 0 = 0, y 0 = 6y 2 , y 0 = y (10 − y ).
Given a differential equation, our goal is to solve it for the unknown
function y . The solution need not be unique, in fact a differential
equation usually has infinitely many solutions!
Compare this with a traditional linear or quadratic equation like
3y − 4 = y ,
y 2 + 2y + 3 = 0
where the solution is a number or at most finitely many numbers.
Math 105 (Section 204)
Differential Equations
2011W T2
1/6
Initial Value Problem
Even though a differential equation has infinitely many solutions in
general, often one is interested in a unique particular solution that
satisfies additional conditions called initial conditions or boundary
conditions. Such conditions specify the values of a solution or a
certain number of its derivatives at some specified value of x.
Math 105 (Section 204)
Differential Equations
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Initial Value Problem
Even though a differential equation has infinitely many solutions in
general, often one is interested in a unique particular solution that
satisfies additional conditions called initial conditions or boundary
conditions. Such conditions specify the values of a solution or a
certain number of its derivatives at some specified value of x.
For example, consider the differential equation y 0 (t) = 2t with initial
condition y (0) = 1. The general solution of the equation is
y (t) = t 2 + C , where C is an arbitrary constant. Among this
infinitely many solutions in this family, only y (t) = t 2 + 1 satisfies the
prescribed initial condition.
Math 105 (Section 204)
Differential Equations
2011W T2
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Initial Value Problem
Even though a differential equation has infinitely many solutions in
general, often one is interested in a unique particular solution that
satisfies additional conditions called initial conditions or boundary
conditions. Such conditions specify the values of a solution or a
certain number of its derivatives at some specified value of x.
For example, consider the differential equation y 0 (t) = 2t with initial
condition y (0) = 1. The general solution of the equation is
y (t) = t 2 + C , where C is an arbitrary constant. Among this
infinitely many solutions in this family, only y (t) = t 2 + 1 satisfies the
prescribed initial condition.
The problem of determining a solution of a differential equation that
satisfies given initial conditions is called an initial value problem.
Math 105 (Section 204)
Differential Equations
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Order of a differential equation
Definition
The order of a differential equation is n if the higher-order derivative that
occurs in the differential equation in n.
Math 105 (Section 204)
Differential Equations
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Order of a differential equation
Definition
The order of a differential equation is n if the higher-order derivative that
occurs in the differential equation in n.
In the previous slide, example 1 is of second order, while examples 2
and 3 are of first order.
We will only be concerned with a special subclass of first order
differential equations known as separable equations.
A first-order differential equation is separable if it can be written in
the form
F (y )dy = G (x)dx,
where the variables x and y have been separated. Note that the left
hand side depends only on y and the right hand side only on x.
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Differential Equations
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Which of these equations is not separable?
A.
dx
t +1
=
dt
tx
B.
y 0 = y 2 − e 3t y 2
C.
t 2y 2
dy
= 3
dt
t y +8
D.
ln x
y0 = √
xy
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Differential Equations
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Method of separation of variables
Solve the initial value problem
3(sec x) y 0 (x) + y 4 = 0,
Math 105 (Section 204)
Differential Equations
y
π 2
1
= .
2
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Method of separation of variables
Solve the initial value problem
3(sec x) y 0 (x) + y 4 = 0,
A. y =
1
2
y
π 2
1
= .
2
sin x
1
B. y = (sin x + 7)− 3
C. y = tan( x2 ) −
1
2
D. y = x/π
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Differential Equations
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Setting up a differential equation
Some homeowner insurance policies include automatic inflation coverage
based on the construction cost index (CCI) as published by the National
Department of Commerce. Each year the property insurance coverage is
increased by an amount based on the change in CCI. Let f (t) be the CCI
at time t years since January 1, 2010 and let f (0) = 100. Suppose the
construction cost index is rising at a rate proportional to the CCI and the
index was e 4 times its starting value on January 1, 2012. Construct and
solve the differential equation satisfied by f (t). Then determine when the
CCI will reach e 7 times its starting value.
A. never
B. in 7 years
C. in 3.5 years
D. in 5 years
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Differential Equations
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