Differential Equation

1. (x - ylny + ylnx) dx + x(lny - lnx) dy= 0
2. (x csc y/x - y) dx + xdy=0
3. (x^2 + 2xy - 4y^2) dx - ( x^2 - 8xy - 4 y^2)=0
4. x^y ' = 4x^2 + 7xy + 2 y^2

eliminate the arbitrary constant:

y=c₁e^x + c₂xe^x

y'=c₁e^x + c₂(xe^x + e^x)
y''=c₁e^x + c₂(xe^x + 2e^x)

by comparing eq.1 and eq,2 = eq,4
by comparing eq.2 and eq.3 = eq.5

by comparing eq.4 and eq.5 = y'' + 2y - 3y'

am i doing it right??

At 1:00pm., a thermometer reading 70F is taken outside where the air temperature is -10F (ten below zero). At 1:02pm., the reading is 26F. At 1:05pm., the thermometer is taken back indoors, where the air is at 70F. What is the temperature reading at 1:09pm?

1. (4D + 1)^4 y = 0.
2. (6D − 5)^3 y = 0.

1. (2xD2 + 1)(3xD + 2)
2. (xD − x)(xD2 − 2)

(sin x)y''' − 3xy'' + 2y = tan x

Let
f1(x) = 1 + x^3 for x ≤ 0 ,1 for x ≥ 0
f2(x)= 1 for x ≤ 0, 1 + x^3 for x ≥ 0
f3(x)= 3 + x^3 for all x.

show that
a) f, f' , f", are continous for all x for each f1, f2, f3.

The given two-parameter family is a solution of the indicated differential equation on the
interval (−∞,∞). Determine whether a member of the family can be found that satisfies the
initial conditions.
y = c1e^x cos x + c2e^x sin x; y" − 2y' + 2y = 0,
i. y(0) = 1, y'(π) = 0
ii. y(0) = 1, y(π) = −1

substitution suggested by equation
1. (3siny - 5x)dx + 2x^2cotydy = 0
2. (ke^2v - u)du = 2e^2v(e^2v + ku)dv = 0

does anyone have a copy of it's solution manual?