For the following ordinary differential equations identify t

For the following ordinary differential equations, identify their dependent variables, independent variables, their orders and comment if they are linear or nonlinear and homogeneous or nonhomogeneous: x^2 d^2y/dx^2 + dy/dx + (x^2 - n^2)y + tan(x) = 0 e^-8z d^5w/dz^5 + sin(w) - z^2 = 0 Evaluate integral e^x cosxdx Consider y\' + (x + 3 )y2 = 0. Find its general solution Show steps of derivation Check your answer by substitution y and y\' into the equation. Solve the following initial value problem and verify your solution by substitution: y^2(t+1)(t+2)dy/dt+t+3=0 with y(0) = 0

Solution

1. (b)

Consider the integration

Integrating by parts we get,

is the integrating constant.

Therefore

2. (a)

Consider the differential equation

The given differential equation can be written as

Integrating both side we get,

is the integrating constant.

Therefore the general solution of the differential equation

is

again,

Derivative both sides with respect to we get

Which is verified.