Differential Equations Answers

How to find uniqueness of solution using characteristics curve and initial condition

Find the solution to 2cos(x)y'(x)=2cos^2(x)-sin^2(x)+y^2 if y(x)=sin x denotes a particular solution

Solve the Cauchys Linear Differential Equations :

x^2.d^2y/dx^2 + x.dy/dx - y = x^3.e^4

(x²+2xy-7x)dx+(x²+2y²-3y)dy=0

A cup of water is cooling. Its initial temperature is 100𝑜𝐶. After 3 minutes its temperature is 80𝑜𝐶. The temperature 𝑇 of the water, measured in ℃, is modelled by 𝑑𝑇 𝑑𝑡 = −𝑘(𝑇 − 25) where 𝑡 is the time elapsed in minutes. (i) Show that 𝑇 − 𝐴𝑒 −𝑘𝑡 − 25 = 0, where 𝐴 and 𝑘 are appropriate constants, (ii) Find the temperature of the water after 5 minutes.

Solve the differential equation 𝑑𝑦 𝑑𝑥 + 2𝑦 tan 𝑥 = sin 𝑥 , 𝑦 ( 𝜋 3 ) = 0.

A large tank is filled to capacity with 700 gallons of pure water. Brine containing 2 pounds of salt per gallon is pumped into the tank at a rate of 7 gal/min. The well-mixed solution is pumped out at a rate of 14 gals/min. Find the number A(t) of pounds of salt in the tank at time t.

(x³-y)dx+xdy=0

(2+ycosx)dx+cosxdy=0