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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.
find the equation of the orthogonal trajectories of y^2+y=ce^x
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
Solve the differential equation by Bernoulli equation. Show complete solution.
dy/dx + 1/3 y = 1/3 (1 + 3x) y4
Solve the differential equation by substitution suggested by equation. Show complete solution.
(x + 2y - 1)dx + 3(x + 2y)dy = 0
