MAT3700 Assignment 1 2026 | Due 4 May 2026 - Distinction Guaranteed

MAT3700 ASSIGNMENT 1 2026

DUE 4 MAY 2026


Solve the following first-order differential equations, question 1 to 4.

QUESTION 1

1. Solve the differential equation:

dy
cos(y) = 3x 2 sin(x 3 )
dx




QUESTION 1 (Separable ODE)




dy
cos y = 3x2 sin(x3 )
dx
​




Separate variables:

cos y dy = 3x2 sin(x3 ) dx

Integrate both sides:


∫ cos y dy = ∫ 3x2 sin(x3 ) dx

Left side: ∫ cos y dy = sin y + C1 ​




Right side: Let u = x3 , then du = 3x2 dx


∫ 3x2 sin(x3 )dx = ∫ sin u du = − cos u + C2 = − cos(x3 ) + C2
​ ​




Equating:

sin y = − cos(x3 ) + C

where C = C2 − C1 .
​ ​




sin y + cos(x3 ) = C

, QUESTION 2 (Homogeneous ODE)


(x2 + 3y 2 ) dx = 2xy dy

Use substitution y = vx.

Rewrite as:


dy x2 + 3y 2 1 + 3(y/x)2 1 + 3v 2
= = =
dx 2xy 2(y/x) 2v
dy
Substitute y = vx, dx
= v + x dv
dx
:

dv 1 + 3v 2
v+x =
dx 2v

dv 1 + 3v 2 1 + 3v 2 − 2v 2 1 + v2
x = −v = =
dx 2v 2v 2v
Separate variables:

2v dx
dv =
1 + v2 x
Integrate:

2v dx
∫ dv = ∫
1 + v2 x

ln ∣1 + v 2 ∣ = ln ∣x∣ + C

1 + v 2 = Cx