Question 1 Parts of this problem are not related

Question 1

Parts of this problem are not related.

1. Solve and obtain the general solution of the following differential equation.

 xdy — (1 + Y² )dx= 0

2. Find the general solution of the following linear differential equation.

Y^{1 –} 1/t*y = t²

Question 2   Parts of this problem are not related.

Write down the form of the particular solution to the following differential equation. Justify your answer, but DO NOT solve for the coefficients.

y(4) —2y⁽²⁾ —8y = t²e^2t + sin(√2t)

What are the singular points of the following differential equation?

(x+1)²y^” + xy^’+ x²y=0

Classify them as regular or irregular.

Question 3

Consider the differential equation

(xy + Y²)dx - (x2 + xy) dY = 0.

1. Show that this differential equation is not exact.
2. Find an integrating factor to make the given differential equation exact. DO NOT solve the differential equation.

Question 4

Consider the differential equation

Y^{” –} xy’ –y = 0

1. Decide if x₀ = 0 is an ordinary or a singular point. Justify
2. Solve the differential equation by means of an infinite series about the given point x₀. Write down the recurrence relation explicitly.

Question 5

Find the general solution of the initial value problem:

 y" + 3y' + 2y = 8(t-5) + u10(t),  y(0) = 0, y’(0) = ½

Question 6

 In the following, compute the Laplace transform or its inverse as required.

(a) f(t) = ∫₀^t (t — ?)³ sin ? d?                 f(s)=l?{f(t)} =?

(b) f(t) ={5-t if 0≤ 1                                 F(s) = ∫ {f(t)}=

                      —2 if t > 1

Question 7

Find the fundamental set of solutions to the system of differential equations.

 X’(2 —1\  3 —2) x.

b) Find the particular solution satisfying the initial conditions x₁(0) = 2 and x₂(0) = —1.}

c) Convert the above system of differential equations and the given initial conditions into an equivalent single higher order differential equation.

Question 8  a) Determine an interval in which a unique solution to the IVP below is guaranteed to exist.

(2x — 3)y(⁴) + (x + 1)y(²) + (tan x)y = 0, y(2) = 2, y'(2) = —2.

b) Show that y₁ =and y₂ = ²√x³ are solutions of 3yy" — (y¹)2 = 0, but that their sum y = y₁ + y₂ is not a solution. Does this violate the principle of superposition? Explain clearly.

Question 9

The slope field for the differential equation

Dy/dx = Y(Y — 3)(y — 2), 

and four solutions satisfying different initial conditions are shown in the figure below:

a) If y = f(x) is a solution to the differential equation such that f is an increasing function for all x, what are all the possible values of the initial conditions y₀ = f(0)?

b) Sketch the phase diagram for the given differential equation. Find and classify the equilibrium points as stable, unstable or semi-stable.