Differential Equations Answers

Given the partial differential equation 𝑈𝑡 + 𝑈𝑥𝑥 = 0, 0 < 𝑥 < 𝜋,𝑡 > 0. Use the method of separation of variables to solve the given equation with the following conditions. 𝑈𝑥 (0,𝑡) = 0,𝑡 > 0

𝑈𝑥 (𝜋,𝑡) = 0,𝑡 > 0 𝑈(𝑥, 0) = 𝑥(𝜋 − 𝑥), 0 < 𝑥 < 𝜋

The temperature of air is 30°C, and the substance cools from 100°C to 70°C in 15 minutes. Find the temperature after 20 min.  

solve dx/2x(y+z^2) = dy/y(2y+z^2) = dz/z^3

find the general solution of the equation 2x(y+z^2)p+y(2y+z^2)q=z^3. and deduce that

yz(z^2+yz-2y)=x^2 is a solution

y''+4y=cos^2(x)

(𝟐𝒙𝒚 − 𝒚^ 𝟐 + 𝒚)𝒅𝒙 + (𝟑𝒙^𝟐 − 𝟒𝒙𝒚 + 𝟑𝒙)𝒅𝒚 = 𝟎

(x+y-4)dx-(3x-y-4)dy=0 when x=4 and y=7.

z=e^(x-2y) x=cost y=sint dz/dt=?

The difference equation generated by y=a/x+b

dy/dx= −(x^2/y^2)+1/[3y^2(x3+ y3)^2]