Cs130 day08

First-Order Differential Equations

Definitions

CS 130: Mathematical Methods in Computer Science Ordinary Differential Equations Nestine Hope S. Hernandez Algorithms and Complexity Laboratory Department of Computer Science University of the Philippines, Diliman nshernandez@dcs.upd.edu.ph Day 8

First-Order Differential Equations

Definitions

Ordinary Differential Equations

Definitions

First-Order Differential Equations Separable Equations Homogeneous Linear First-Order ODE Non-homogeneous Linear First-Order ODE

First-Order Differential Equations

Definitions

Ordinary Differential Equations

Definitions

First-Order Differential Equations Separable Equations Homogeneous Linear First-Order ODE Non-homogeneous Linear First-Order ODE

Definitions

First-Order Differential Equations

A differential equation is an equation relating an unknown function to one or more of its derivatives.

First-Order Differential Equations

Definitions

A differential equation is an equation relating an unknown function to one or more of its derivatives.

Example 1.

2.

dy dx = cos x d2 y dx2 + ky = 2 2

0

3. (x + y )dx − 2xydy = 0

∂2u 2 ∂2u 4. ∂u = h + 2 2 ∂t ∂x ∂y 2

d i di 5. L dt 2 + R dt +

1 Ci

= Eω cos ωt

First-Order Differential Equations

Definitions

A differential equation is an equation relating an unknown function to one or more of its derivatives.

Example 1.

2.

dy dx = cos x d2 y dx2 + ky = 2 2

0

3. (x + y )dx − 2xydy = 0

∂2u 2 ∂2u 4. ∂u = h + 2 2 ∂t ∂x ∂y 2

d i di 5. L dt 2 + R dt +

1 Ci

= Eω cos ωt

• independent variable • dependent variable • parameter

————————————

First-Order Differential Equations

Definitions

A differential equation is an equation relating an unknown function to one or more of its derivatives.

Example 1.

2.

dy dx = cos x d2 y dx2 + ky = 2 2

0

3. (x + y )dx − 2xydy = 0

∂2u 2 ∂2u 4. ∂u = h + 2 2 ∂t ∂x ∂y 2

d i di 5. L dt 2 + R dt +

1 Ci

= Eω cos ωt

• independent variable • dependent variable • parameter

———————————— • ordinary differential equations • partial differential equations

Definitions

First-Order Differential Equations

The order of a differential equation is the order of the highest-ordered derivative appearing in the equation.

First-Order Differential Equations

Definitions

The order of a differential equation is the order of the highest-ordered derivative appearing in the equation.

Example :

d2 y dx2

dy 3 + 2b( dx ) +y =0

First-Order Differential Equations

Definitions

The order of a differential equation is the order of the highest-ordered derivative appearing in the equation.

Example :

d2 y dx2

dy 3 + 2b( dx ) +y =0

A solution of a differential equation is a differentiable function that satisfies the equation on some interval (a, b) of values for the independent variable.

First-Order Differential Equations

Definitions

The order of a differential equation is the order of the highest-ordered derivative appearing in the equation.

Example :

d2 y dx2

dy 3 + 2b( dx ) +y =0

A solution of a differential equation is a differentiable function that satisfies the equation on some interval (a, b) of values for the independent variable.

Example y = e2x is a solution of the equation

d2 y dx2

+

dy dx

− 6y = 0.

First-Order Differential Equations

Definitions

Ordinary Differential Equations

Definitions

First-Order Differential Equations Separable Equations Homogeneous Linear First-Order ODE Non-homogeneous Linear First-Order ODE

Definitions

First-Order Differential Equations

Differential equations of order one may be written as

Definitions

First-Order Differential Equations

Differential equations of order one may be written as

dy dx

= f (x, y).

Definitions

First-Order Differential Equations

Differential equations of order one may be written as We can write the equation in the form

dy dx

= f (x, y).

Definitions

First-Order Differential Equations

Differential equations of order one may be written as

dy dx

= f (x, y).

We can write the equation in the form M (x, y)dx + N (x, y)dy = 0.

First-Order Differential Equations

Definitions

Differential equations of order one may be written as

dy dx

= f (x, y).

We can write the equation in the form M (x, y)dx + N (x, y)dy = 0.

Example dy dx

= 1 − xy + y 2

First-Order Differential Equations

Definitions

Differential equations of order one may be written as

dy dx

= f (x, y).

We can write the equation in the form M (x, y)dx + N (x, y)dy = 0.

Example dy dx

= 1 − xy + y 2

If it happens that M is a function of x only and N is a function of y only, then M (x)dx + N (y)dy = 0. Such an equation is said to be separable.

First-Order Differential Equations

Definitions

Separable Equations M (x)dx + N (y)dy = 0 A separable equation can be solved by integrating the functions M and N .

First-Order Differential Equations

Definitions

Separable Equations M (x)dx + N (y)dy = 0 A separable equation can be solved by integrating the functions M and N .

Example Solve the following equations: 1.

dy dx

=

x2 1−y 2

First-Order Differential Equations

Definitions

Separable Equations M (x)dx + N (y)dy = 0 A separable equation can be solved by integrating the functions M and N .

Example Solve the following equations: 1.

dy dx

=

x2 1−y 2

2.

dy dx

=

4x−x3 4+y 3

First-Order Differential Equations

Definitions

Separable Equations M (x)dx + N (y)dy = 0 A separable equation can be solved by integrating the functions M and N .

Example Solve the following equations: 1.

dy dx

=

x2 1−y 2

2.

dy dx

=

4x−x3 4+y 3

3.

dy dx

=

(y−3) cos x 1+2y 2

First-Order Differential Equations

Definitions

Separable Equations M (x)dx + N (y)dy = 0 A separable equation can be solved by integrating the functions M and N .

Example Solve the following equations: 1.

dy dx

=

x2 1−y 2

2.

dy dx

=

4x−x3 4+y 3

3.

dy dx

=

(y−3) cos x 1+2y 2

4.

dy dx

=

3x2 +4x+2 2(y−1)

where y(0) = −1

First-Order Differential Equations

Definitions

Exercises Solve the following equations: 1. xdx + ye−x dy = 0 2. y 0 = (cos 2x)(cos2 y) 3. y 0 = xy 3 (1 + x2 )−1/2 where y(0) = 1

First-Order Differential Equations

Definitions

Homogeneous Linear First-Order ODE Any differential equation of the form a1 (x)y 0 + a0 (x)y = b(x) is a linear first-order differential equation.

First-Order Differential Equations

Definitions

Homogeneous Linear First-Order ODE Any differential equation of the form a1 (x)y 0 + a0 (x)y = b(x) is a linear first-order differential equation.

Any other first-order ODE is said to be nonlinear.

First-Order Differential Equations

Definitions

Homogeneous Linear First-Order ODE Any differential equation of the form a1 (x)y 0 + a0 (x)y = b(x) is a linear first-order differential equation.

Any other first-order ODE is said to be nonlinear.

The simplest linear first-order ODE is of the form y 0 + p(x)y = 0, called a homogeneous linear first-order ODE.

Definitions

Example y 0 + (1 + 3x2 )y = 0

First-Order Differential Equations

First-Order Differential Equations

Definitions

Example y 0 + (1 + 3x2 )y = 0 For any homogeneous fist-order differential equation of the form y 0 + p(x)y = 0, the general solution is Ke竏単 (x) , where P is any antiderivative of p.

First-Order Differential Equations

Definitions

Exercises Solve the following equations: 1. (2 + x2 )y 0 + 2x y = 0 2. y 0 sin 2x + 2y cos 2x = 0 dy dy 3. y + 3x2 dx = x dx

First-Order Differential Equations

Definitions

Non-homogeneous Linear First-Order ODE y 0 + p(x)y = f (x)

First-Order Differential Equations

Definitions

Non-homogeneous Linear First-Order ODE y 0 + p(x)y = f (x)

For any nonhomogeneous linear fist-order differential equation of the form y 0 + p(x)y = f (x), R the general solution is y = e竏単 (x) eP (x) f (x)dx, where P is any antiderivative of p.

First-Order Differential Equations

Definitions

Non-homogeneous Linear First-Order ODE y 0 + p(x)y = f (x)

For any nonhomogeneous linear fist-order differential equation of the form y 0 + p(x)y = f (x), R the general solution is y = e竏単 (x) eP (x) f (x)dx, where P is any antiderivative of p.

Example 1. y 0 + 2y = 4

First-Order Differential Equations

Definitions

Non-homogeneous Linear First-Order ODE y 0 + p(x)y = f (x)

For any nonhomogeneous linear fist-order differential equation of the form y 0 + p(x)y = f (x), R the general solution is y = e竏単 (x) eP (x) f (x)dx, where P is any antiderivative of p.

Example 1. y 0 + 2y = 4 2. y 0 + y tan x = cos x

First-Order Differential Equations

Definitions

Questions?

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