MATH 204

MATH 204

Chapter 1

Introduction to Differential Equations

1.1 Definitions

A differential equation (DE) is any equation which contains the derivatives of one or more dependent variables with respect to one or more independent variables. For example,

d²y

dx² −2xdy

dx +3y=0

is a differential equation in whichyis the dependent variable andxis the independent variable.

Classification of Differential Equations :

Differential equations can be classified by order, type and linearity.

1. Order : The order of a differential equation is the order of the highest derivative in the equation. For example,

dy

dx +y =x

is a first order differential equation, while d²y

dx² +5dy

dx+xy=0 is a second order differential equation.

2. Type : If a differential equation contains only one independent variable, then it is called an ordinary differential equation (ODE). If it contains more than one independent variable, then it is called a partial differential equation (PDE). For example,

dy

dx +5y =e^x, d²y dx² − ^dy

dx +6y =0 are ordinary differential equations, while

∂²y

∂x² +^∂

2y

∂t² =0, ∂²y

∂x² = ^∂

2y

∂t² −2∂y

∂t

are partial differential equations.

3. Linearity :A differential equation is called linear inyif it has the form

an(x)^d

ny

dxⁿ +an−1(x)^d

n−1y

dxⁿ⁻¹ +. . .+a1(x)^dy

dx+a0(x)y =g(x).

If a differential equation is not linear in y, then it is called nonlinear in y. For example, the differential equations

x²dy

dx+3xy=1, d³y

dx³ +xdy

dx −5y=cosx are linear iny, while the differential equations

e^ydy

dx+xy=2, x³d²y

dx² +3y² =0 are nonlinear iny.

Remark. A differential equation is called linear inxif it has the form

an(_y)^d

nx

dyⁿ +_an−1(_y)^d

n−1x

dyⁿ⁻¹ +_{. . .}+_a1(_y)^dx

dy +_a0(_y)_x= _g(_y)_.

If a differential equation is not linear in x, then it is called nonlinear in x. For example, the differential equations

y³dx

dy +2yx=1, yd²x

dy² −3dx

dy +x=lny are linear inx, while the differential equations

3 √

Notations :

1. Ordinary differential equations can be written using the following notations dy

dx, d²y

dx², . . . , y⁰, y⁰⁰, . . . , y, ¨˙ y, . . . . For example, the following equations are equivalent

d²y

dx² −ydy

dx =x, y⁰⁰−yy⁰ =x, y¨−yy˙ =x.

2. Partial differential equations can be written using the following notations

∂y

∂x, ∂²y

∂x², . . . , yx, yxx, . . . . For example, the following equations are equivalent

∂²y

∂x² =₃^∂y

∂t +y, yxx =3yt+y.

Example. Classify the following differential equation based on order, type and linearity : xd²y

dx² −^dy dx

3

+y=0.

Example. Determine whether the following differential equation is linear inxandy: y²−1

dx+xdy =0.

Solutions of Differential Equations :

A solution of a differential equation is any function which satisfies the differential equation. If a solution is given in the formy= f(x), then it is called an explicit solution, otherwise it is called an implicit solution. A solution which contains at least one arbitrary constant is called a general solution, while a solution which is free from arbitrary constants is called a particular solution.

Example. Verify that each given function is a solution of the given differential equation : 1. dy

dx =x√

y, y= ¹ 16x⁴

2. y⁰⁰−2y⁰+y=0, y= xe^x

Example. Verify that the relation x²+y² =9 is an implicit solution of the following differential equation

y⁰ =−^x y·

Initial and Boundary Value Problems :

1. An initial value problem (IVP) is annth order differential equation together withnconditions imposed on the solution at one point. For example,

y⁰ =6x, y(1) = 3

and

y⁰⁰−4y=12x, y(0) =4, y⁰(0) =1 are initial value problems.

2. A boundary value problem (BVP) is annth order differential equation together withnconditions imposed on the solution at more than one point. For example,

y⁰⁰−y⁰+2y =0, y(0) =−2, y(1) = 2

is a boundary value problem.

Exercises 1.1

Q1. Classify the following differential equations based on order, type and linearity : 1. (1−x)y⁰⁰−4xy⁰+5y=cosx

2. d³y dx³ =

r

1+^dy dx

4

Q2. Determine whether the following differential equation is linear inxandy: xdy+ (y+xy−xe^x)dx =0.

Q3. Verify that each given function is a solution of the given differential equation : 1. 2y⁰+y =0, y =e^−x/2

2. dy

dt +20y =24, y = ⁶ 5 −⁶

5e^−20t

Answers 1.1

A1.

1. 2nd order, ODE, linear iny.

2. 3rd order, ODE, nonlinear iny.

A2. Linear iny, nonlinear in x.

A3. Verify.