UNIVERSITY OF MALTAFACULTY OF SCIENCE

DEPARTMENT OF MATHEMATICS

MAT1801 Mathematics for Engineers IProblem Sheet 2

First Order Ordinary Differential Equations

(1) Solve the differential equation

ydy

dx= ex+2y sin x.

Ans. 2ex(sin x− cos x) + e−2y(2y + 1) = C

(2) Solve the differential equation

2x3 dy

dx= y2 + 3xy2,

given that y = 1 when x = 1.

Ans. 4x2 = y(1 + 6x− 3x2)

(3) Solve the differential equation

dy

dx=

x + y − 2

x + y + 2.

Ans. x− y = ln(x + y)2 + C

(4) Solve the differential equation

(2x− 4y − 8)dy

dx= 3x− 5y − 9.

Ans. 3x− 4y − 6 = C(x− y − 1)2

(5) Solve the differential equation

dy

dx=

y

x+ x sin

(y

x

).

Hint: Use the substitution v = yx.

Ans. y = 2x tan−1(Cex)

(6) Solve the differential equation

4x2 dy

dx= 4xy − x2 + y2,

given that when x = 1, y = 2.

Ans. x(x + y)2 = 9(x− y)2

1

(7) Reduce the differential equation

(x + 1)ydy

dx− y2 = x

to a linear form by writing v = y2, and solve it, given that whenx = 0, y = 1.

Ans. y2 = 2x2 + 2x + 1

(8) Solve the differential equation

dy

dx=

(1 + 2ex)y

x + 2ex,

given that y(0) = 1.

Ans. y = 12(x + 2ex)

(9) Solve the given initial value problem using separation of vari-ables:

dy

dx=

2x

x2 + 2y,

given that y(0) = 2.

Ans. y = x2 + 2

(10) Solve the given initial value problem

dy

dx=−1

xy + cos x,

given that y(1) = 0.

Ans. y = sin x + 1x(cos x− cos 1− sin 1)

(11) Convert the following differential equation into two equationsof the form v = y′ and v′ = F (x, y′) and solve for the generalsolution, containing two arbitrary constants:

d2y

dx2+ 2

dy

dx= ex.

Ans. y = K + 13ex − 1

2Ce−2x

(12) Convert the following differential equation into two equationsof the form v(dv/du) = F (u, v), using u = y(x) and v = y′(x).Solve for the general solution, containing two arbitrary con-stants, if possible:

2yd2y

dx2= 1 +

(dy

dx

)2

.

Ans. y = C4x2 + Kx +

(K2+1

C

)

2

(13) Solve the differential equation

2y′ sin x− y cos x = y3 sin x cos x.

Find also the particular solution for which y = −1 when x = 12π.

Ans. y2(C − sin2 x) = 2 sin x, y2(3− sin2 x) = 2 sin x

(14) The equation of motion for the velocity v(t) of a rocket in aconstant gravitational field g is

(M − at)v′(t)− ab = −g(M − at).

In this equation, M is the initial rocket mass in grams and therocket loses gass at a grams per second at constant velocity bcentimeters per second relative to the rocket (all constants arepositive). Solve the equation for the position x(t) and velocityv(t).

Hint: Note that v(t) =d

dtx(t), and the initial conditions are

v(0) = 0 and x(0) = Re (radius of the earth). The answersinvolve the reciprocal and logarithm of M − at.

Ans. x = Re + bt− gt2

2+ b

a(M − at) ln

∣∣1− atM

∣∣

3