BASIC CONCEPTS

CHAPTER 1Introduction to Differential Equations

Differential Equations

BASIC CONCEPTS

Differential Equations

• Definition: A differential equation is an equation thatcontains a function and one or more of its derivatives. Ifthe function has only one independent variable, then it isan ordinary differential equation. Otherwise, it is apartial differential equation.

• The following are examples of differential equations:

(a)∂2u∂x2 +

∂2u∂y2 = 0

(b) (x2 + y2)dx − 2xydy = 0

(c)d3xdy3

+ xdxdy− 4xy = 0

(d)∂u∂t

= h2(∂2u∂x2 +

∂2u∂y2

)

BASIC CONCEPTS

Differential Equations

• Definition: A differential equation is an equation thatcontains a function and one or more of its derivatives. Ifthe function has only one independent variable, then it isan ordinary differential equation. Otherwise, it is apartial differential equation.

• The following are examples of differential equations:

(a)∂2u∂x2 +

∂2u∂y2 = 0

(b) (x2 + y2)dx − 2xydy = 0

(c)d3xdy3

+ xdxdy− 4xy = 0

(d)∂u∂t

= h2(∂2u∂x2 +

∂2u∂y2

)

BASIC CONCEPTS

Differential Equations

• Definition: A differential equation is an equation thatcontains a function and one or more of its derivatives. Ifthe function has only one independent variable, then it isan ordinary differential equation. Otherwise, it is apartial differential equation.

• The following are examples of differential equations:

(a)∂2u∂x2 +

∂2u∂y2 = 0

(b) (x2 + y2)dx − 2xydy = 0

(c)d3xdy3

+ xdxdy− 4xy = 0

(d)∂u∂t

= h2(∂2u∂x2 +

∂2u∂y2

)

BASIC CONCEPTS

Differential Equations

• Definition: A differential equation is an equation thatcontains a function and one or more of its derivatives. Ifthe function has only one independent variable, then it isan ordinary differential equation. Otherwise, it is apartial differential equation.

• The following are examples of differential equations:

(a)∂2u∂x2 +

∂2u∂y2 = 0

(b) (x2 + y2)dx − 2xydy = 0

(c)d3xdy3

+ xdxdy− 4xy = 0

(d)∂u∂t

= h2(∂2u∂x2 +

∂2u∂y2

)

BASIC CONCEPTS

Differential Equations

• Definition: A differential equation is an equation thatcontains a function and one or more of its derivatives. Ifthe function has only one independent variable, then it isan ordinary differential equation. Otherwise, it is apartial differential equation.

• The following are examples of differential equations:

(a)∂2u∂x2 +

∂2u∂y2 = 0

(b) (x2 + y2)dx − 2xydy = 0

(c)d3xdy3

+ xdxdy− 4xy = 0

(d)∂u∂t

= h2(∂2u∂x2 +

∂2u∂y2

)

BASIC CONCEPTS

Differential Equations

• Definition: A differential equation is an equation thatcontains a function and one or more of its derivatives. Ifthe function has only one independent variable, then it isan ordinary differential equation. Otherwise, it is apartial differential equation.

• The following are examples of differential equations:

(a)∂2u∂x2 +

∂2u∂y2 = 0

(b) (x2 + y2)dx − 2xydy = 0

(c)d3xdy3

+ xdxdy− 4xy = 0

(d)∂u∂t

= h2(∂2u∂x2 +

∂2u∂y2

)

BASIC CONCEPTS

Order and Degree

• Definition: The order of a differential equation is the orderof the highest ordered derivative that appears in the givenequation. The degree of a differential equation is thedegree of the highest ordered derivative treated as avariable.

• Examples:

(a)∂2u∂x2 +

∂2u∂y2 = 0 is of order 2 and degree 1

(b) (x2 + y2)dx − 2xydy = 0 is of order 1 and degree 1

(c)(

d3xdy3

)2

+ xdxdy− 4xy = 0 is of order 3 and degree 2

(d)∂u∂t

= h2(∂2u∂x2 +

∂2u∂y2

)3

is of order 2 and degree 3

BASIC CONCEPTS

Order and Degree

• Definition: The order of a differential equation is the orderof the highest ordered derivative that appears in the givenequation. The degree of a differential equation is thedegree of the highest ordered derivative treated as avariable.

• Examples:

(a)∂2u∂x2 +

∂2u∂y2 = 0 is of order 2 and degree 1

(b) (x2 + y2)dx − 2xydy = 0 is of order 1 and degree 1

(c)(

d3xdy3

)2

+ xdxdy− 4xy = 0 is of order 3 and degree 2

(d)∂u∂t

= h2(∂2u∂x2 +

∂2u∂y2

)3

is of order 2 and degree 3

BASIC CONCEPTS

Order and Degree

• Definition: The order of a differential equation is the orderof the highest ordered derivative that appears in the givenequation. The degree of a differential equation is thedegree of the highest ordered derivative treated as avariable.

• Examples:

(a)∂2u∂x2 +

∂2u∂y2 = 0 is of order 2 and degree 1

(b) (x2 + y2)dx − 2xydy = 0 is of order 1 and degree 1

(c)(

d3xdy3

)2

+ xdxdy− 4xy = 0 is of order 3 and degree 2

(d)∂u∂t

= h2(∂2u∂x2 +

∂2u∂y2

)3

is of order 2 and degree 3

BASIC CONCEPTS

Order and Degree

• Definition: The order of a differential equation is the orderof the highest ordered derivative that appears in the givenequation. The degree of a differential equation is thedegree of the highest ordered derivative treated as avariable.

• Examples:

(a)∂2u∂x2 +

∂2u∂y2 = 0 is of order 2 and degree 1

(b) (x2 + y2)dx − 2xydy = 0 is of order 1 and degree 1

(c)(

d3xdy3

)2

+ xdxdy− 4xy = 0 is of order 3 and degree 2

(d)∂u∂t

= h2(∂2u∂x2 +

∂2u∂y2

)3

is of order 2 and degree 3

BASIC CONCEPTS

Order and Degree

• Definition: The order of a differential equation is the orderof the highest ordered derivative that appears in the givenequation. The degree of a differential equation is thedegree of the highest ordered derivative treated as avariable.

• Examples:

(a)∂2u∂x2 +

∂2u∂y2 = 0 is of order 2 and degree 1

(b) (x2 + y2)dx − 2xydy = 0 is of order 1 and degree 1

(c)(

d3xdy3

)2

+ xdxdy− 4xy = 0 is of order 3 and degree 2

(d)∂u∂t

= h2(∂2u∂x2 +

∂2u∂y2

)3

is of order 2 and degree 3

BASIC CONCEPTS

Order and Degree

• Definition: The order of a differential equation is the orderof the highest ordered derivative that appears in the givenequation. The degree of a differential equation is thedegree of the highest ordered derivative treated as avariable.

• Examples:

(a)∂2u∂x2 +

∂2u∂y2 = 0 is of order 2 and degree 1

(b) (x2 + y2)dx − 2xydy = 0 is of order 1 and degree 1

(c)(

d3xdy3

)2

+ xdxdy− 4xy = 0 is of order 3 and degree 2

(d)∂u∂t

= h2(∂2u∂x2 +

∂2u∂y2

)3

is of order 2 and degree 3

BASIC CONCEPTS

Solution of a Differential Equation

• Definition: A solution of a differential equation is afunction defined explicitly or implicitly by an equation thatsatisfies the given equation. The general solutionrepresents all the possible solutions of the given equation,while a particular solution is any one of the possiblesolutions of a given differential equation.

• Examples:

(a)dydx

= y , y = Cex where C is an arbitrary constant

(b)dydx

= 3ex , y = 3ex + C where C is an arbitrary constant

(c) y (3) − 3y ′ + 2y = 0, y = e−2x

(d)dydx

=−(x + 1)

y − 3, (x + 1)2 + (y − 3)2 = c2, where c is an

arbitrary constant.

(e)d2ydt2 + k2y = 0, y = sin kt , where k is a constant

BASIC CONCEPTS

Solution of a Differential Equation

• Definition: A solution of a differential equation is afunction defined explicitly or implicitly by an equation thatsatisfies the given equation. The general solutionrepresents all the possible solutions of the given equation,while a particular solution is any one of the possiblesolutions of a given differential equation.

• Examples:

(a)dydx

= y , y = Cex where C is an arbitrary constant

(b)dydx

= 3ex , y = 3ex + C where C is an arbitrary constant

(c) y (3) − 3y ′ + 2y = 0, y = e−2x

(d)dydx

=−(x + 1)

y − 3, (x + 1)2 + (y − 3)2 = c2, where c is an

arbitrary constant.

(e)d2ydt2 + k2y = 0, y = sin kt , where k is a constant

BASIC CONCEPTS

Solution of a Differential Equation

• Definition: A solution of a differential equation is afunction defined explicitly or implicitly by an equation thatsatisfies the given equation. The general solutionrepresents all the possible solutions of the given equation,while a particular solution is any one of the possiblesolutions of a given differential equation.

• Examples:

(a)dydx

= y , y = Cex where C is an arbitrary constant

(b)dydx

= 3ex , y = 3ex + C where C is an arbitrary constant

(c) y (3) − 3y ′ + 2y = 0, y = e−2x

(d)dydx

=−(x + 1)

y − 3, (x + 1)2 + (y − 3)2 = c2, where c is an

arbitrary constant.

(e)d2ydt2 + k2y = 0, y = sin kt , where k is a constant

BASIC CONCEPTS

Solution of a Differential Equation

• Definition: A solution of a differential equation is afunction defined explicitly or implicitly by an equation thatsatisfies the given equation. The general solutionrepresents all the possible solutions of the given equation,while a particular solution is any one of the possiblesolutions of a given differential equation.

• Examples:

(a)dydx

= y , y = Cex where C is an arbitrary constant

(b)dydx

= 3ex , y = 3ex + C where C is an arbitrary constant

(c) y (3) − 3y ′ + 2y = 0, y = e−2x

(d)dydx

=−(x + 1)

y − 3, (x + 1)2 + (y − 3)2 = c2, where c is an

arbitrary constant.

(e)d2ydt2 + k2y = 0, y = sin kt , where k is a constant

BASIC CONCEPTS

Solution of a Differential Equation

• Definition: A solution of a differential equation is afunction defined explicitly or implicitly by an equation thatsatisfies the given equation. The general solutionrepresents all the possible solutions of the given equation,while a particular solution is any one of the possiblesolutions of a given differential equation.

• Examples:

(a)dydx

= y , y = Cex where C is an arbitrary constant

(b)dydx

= 3ex , y = 3ex + C where C is an arbitrary constant

(c) y (3) − 3y ′ + 2y = 0, y = e−2x

(d)dydx

=−(x + 1)

y − 3, (x + 1)2 + (y − 3)2 = c2, where c is an

arbitrary constant.

(e)d2ydt2 + k2y = 0, y = sin kt , where k is a constant

BASIC CONCEPTS

Solution of a Differential Equation

• Definition: A solution of a differential equation is afunction defined explicitly or implicitly by an equation thatsatisfies the given equation. The general solutionrepresents all the possible solutions of the given equation,while a particular solution is any one of the possiblesolutions of a given differential equation.

• Examples:

(a)dydx

= y , y = Cex where C is an arbitrary constant

(b)dydx

= 3ex , y = 3ex + C where C is an arbitrary constant

(c) y (3) − 3y ′ + 2y = 0, y = e−2x

(d)dydx

=−(x + 1)

y − 3, (x + 1)2 + (y − 3)2 = c2, where c is an

arbitrary constant.

(e)d2ydt2 + k2y = 0, y = sin kt , where k is a constant

BASIC CONCEPTS

Existence and Uniqueness Theorem

The existence of a particular solution satisfying initial conditionsof the form y(x0) = y0 is guaranteed by the following theorem:Existence and Uniqueness Theorem: Consider a first orderequation of the form

dydx

= f (x , y)

and let T be the rectangular region described by T = { (x , y) ∈R2 | |x − x0| ≤ a, |y − y0| ≤ b, a,b are positive constants }. If fand fy are continuous in T , then there exists a positive numberh and a function y = y(x) such that(a) y = y(x) is a solution of the given equation satisfying

y(x0) = y0; and(b) y = y(x) is unique on the interval |x − x0| ≤ h.

BASIC CONCEPTS

Exercises

(1) Verify that the given function is a solution of the givendifferential equation.(a) y (3) − 3y ′ + 2y = 0, y = e−2x

(b)d2ydt2 + k2y = 0, y = sin kt , where k is a constant

(2) Use antiderivatives to obtain a general or a particularsolution to each of the following equations:

(a)dydx

= x3 + 2x

(b)dydx

= 4 cos 2x

(c)dydx

= 3ex , y = 6 x = 0

(d)dydx

= 4y , y = 3 whenx = 0