Understanding Engineering Mathematics

From

dy

dx

=x^2 we have

y=

∫

x^2 dx+C=

x^3

3

+C

which is the required general solution.

(a) Ify=1whenx=0thenwehave:

y= 1 = 0 +C

soC=1 and the required particular solution isy=

x^3

3

+ 1

(b)y=0whenx=1gives

0 =

13

3

+C=

1

3

+C

soC=−^13 and the required particular solution is

y=

x^3

3

−

1

3

=

x^3 − 1

3

Exercises on 15.2

1. State the order of the following differential equations. Which are nonlinear?

(i)

dy

dx

=ex+ 1 (ii)

d^2 y

dx^2

− 9 y= 0

(iii) y

d^2 y

dx^2

+cosx=0(iv)

dy

dx

d^3 y

dx^3

+ 2 y^2 = 1

(v)

d^2 y

dx^2

− 4

dy

dx

+ 3 y= 3 x+ 2

2. Verify that the following functions are each solutions of one of the equations in Q1,
   and match the solution to its equation.

(a) 2e^3 x (b) ex+x+ 2

(c) e^3 x+x+ 2

3. Find the general solution of Q1(i) and the particular solution that satisfiesy( 0 )=1.

Answers

1. (i) 1 (ii) 2 (iii) 2 (iv) 3

(v) 2

(iii) and (iv) are nonlinear.

2. (a) 1 (ii) (b) 1 (i) (c) 1 (v)


3. y=ex+x+C;y=ex+x