MAT1512 ASSIGNMENT 3 2025

MAT1512
ASSIGNMENT 3
2025

, QUESTION 1


a).


y 2 e2x = 3y + x 2
d 2 2x d
[y e ] = [3y + x 2 ]
dx dx
d 2 2x d d
[y e ] = [3y] + [x 2 ]
dx dx dx
d 2 d d d
e2x [y ] + y 2 [e2x ] = [3y] + [x 2 ]
dx dx dx dx
dy dy
e2x (2y) + y 2 (2e2x ) = 3 + 2x
dx dx
dy dy
2ye2x + 2y 2 e2x = 3 + 2x
dx dx
dy dy
2ye2x −3 = 2x − 2y 2 e2x
dx dx
dy
(2ye2x − 3) = 2x − 2y 2 e2x
dx
dy 2x − 2y 2 e2x
=
dx 2ye2x − 3


Let equation of tangent: y = m1 x + C




dy
m1 =
dx
2x − 2y 2 e2x
m1 =
2ye2x − 3


At point (0,3) ∶


2x − 2y 2 e2x
m1 = ∴ y = 3 when x = 0
2ye2x − 3

, 2(0) − 2(3)2 e2(0)
m1 =
2(3)e2(0) − 3
0 − 18
m1 =
6−3
−18
m1 =
3
m1 = −6


y = m1 x + C ∴ y = 3 when x = 0
3 = (−6)(0) + C
C=3


Equation of the tangent ∶


y = m1 x + C
y = −6x + 3


Let ∶ m2 = gradient of normal line at point (0,3)


m1 m2 = −1
1
m2 = −
m1
1
m2 = −
−6
1
m2 =
6


Let equation of normal line: y = m2 x + K


y = m2 x + K ∴ y = 3 when x = 0
1
3 = ( ) (0) + K
6
K=3