MAT3700 – UNISA Assessment 1
Solutions
Question 1
Given:
cos(y) dy/dx = 3x² sin(x³)
Separating variables:
cos(y) dy = 3x² sin(x³) dx
Integrating both sides:
∫ cos(y) dy = ∫ 3x² sin(x³) dx
sin(y) = -cos(x³) + C
Question 2
Given:
(x² + 3y²)dx = 2xy dy
Let y = vx, then dy/dx = v + x dv/dx
Substituting:
x² + 3v²x² = 2x(vx)(v + x dv/dx)
Simplifying:
1 + 3v² = 2v(v + x dv/dx)
1 + v² = 2vx dv/dx
Separating variables:
dv/(1 + v²) = dx/(2x)
Integrating:
tan⁻¹(v) = (1/2) ln|x| + C
Solutions
Question 1
Given:
cos(y) dy/dx = 3x² sin(x³)
Separating variables:
cos(y) dy = 3x² sin(x³) dx
Integrating both sides:
∫ cos(y) dy = ∫ 3x² sin(x³) dx
sin(y) = -cos(x³) + C
Question 2
Given:
(x² + 3y²)dx = 2xy dy
Let y = vx, then dy/dx = v + x dv/dx
Substituting:
x² + 3v²x² = 2x(vx)(v + x dv/dx)
Simplifying:
1 + 3v² = 2v(v + x dv/dx)
1 + v² = 2vx dv/dx
Separating variables:
dv/(1 + v²) = dx/(2x)
Integrating:
tan⁻¹(v) = (1/2) ln|x| + C
