Differential Equations Answers

Solve the equation (Bernoulli Differential Equation)

(2y^3 - x^3) dx + 3x^2y dx = 0

(x^2 - 1)dy - (y^2 - 1)dx

ut =9uxx

u(0,t)=u(2π,t) 0, t > 0

u(x,0) = {x , 0 < x < π

{- 3x, π < x < 2π

ut =50uxx

u(0,t) = u(π,t) = 0, t > 0

u(x,0) = { x , 0 < x < π / 2

{ 4 , π / 2 < x < π

prove that u(x,y)=x²y is an integrating factor of the equation

(3y+4xy²)dx+(2x+3x²y)dy=0

Hence solve the equation

Find an integrating factor of the form yⁿ for the equation

(y²+2xy)dx-x²dy=0. Hence solve the equation

Find the general solution of each of the following

i) (2xsiny+y³e^x)dx +(x²cosy+3y²e^x)dy=0

ii) (ysec²x+secxtanx)dx + (tanx+2y)dy=0

iii)(ye^x+2e^x+y²)dx+(e^x+2xy)dy=0 y(0)=6

iv)(2xcosy+3x²y)dx+(x³-x²siny-y)dy=0 y(1)=3

Solve the following differential equation (3 marks)

d

2

y

dx2

− 6

dy

dx + 9y = x

2

e

3x

using the method of undetermined coefficients.