. Classify the following partial differential equations:(i) 𝜕2𝑢𝜕𝑥2 + 3 𝜕2𝑢𝜕𝑦2 = 𝜕

The equationsin(xy)(dydx)3+2x=0sin⁡(𝑥𝑦)(𝑑𝑦𝑑𝑥)3+2𝑥=0is a Question 38Select one:a.None of the othersb.Separable differential equationc.Third order differential equationd.Second order differential equatione.First order differential equation

Using the method of separation of variables, solve the following partial differentialequations:(i) 𝜕𝑢𝜕𝑥 = 2 𝜕𝑢𝜕𝑡 + 𝑢; given that 𝑢(𝑥, 0) = 6𝑒−3𝑥.(ii) 4 𝜕𝑢𝜕𝑡 + 𝜕𝑢𝜕𝑥 = 3𝑢; given that 𝑢(𝑥, 0) = 3𝑒−𝑥 − 𝑒−5𝑥

A system described by the following differential equation   + 3  +2𝑦 = x(𝑡) is initially at rest. For input x(𝑡) = 2𝑢(𝑡), the output y(t) isSelect one:a. (1 + 2𝑒 −𝑡 − 2𝑒 −2𝑡 ) 𝑢(𝑡)b. (0.5 + 2𝑒 −𝑡 + 2𝑒 −2𝑡 ) 𝑢(𝑡)c. (0.5 + 𝑒 −𝑡 + 1.5𝑒 −2 ) 𝑢(𝑡)d. (1 − 2𝑒 −𝑡 + 𝑒 −2 ) 𝑢(𝑡)

A system described by the following differential equation   + 3  +2𝑦 = x(𝑡) is initially at rest. For input x(𝑡) = 2𝑢(𝑡), the output y(t) isSelect one:a. (0.5 + 2𝑒 −𝑡 + 2𝑒 −2𝑡 ) 𝑢(𝑡)b. (0.5 + 𝑒 −𝑡 + 1.5𝑒 −2 ) 𝑢(𝑡)c. (1 + 2𝑒 −𝑡 − 2𝑒 −2𝑡 ) 𝑢(𝑡)d. (1 − 2𝑒 −𝑡 + 𝑒 −2 ) 𝑢(𝑡)

A system described by the following differential equation   + 3  +2𝑦 = x(𝑡) is initially at rest. For input x(𝑡) = 2𝑢(𝑡), the output y(t) isSelect one:a. (1 − 2𝑒 −𝑡 + 𝑒 −2 ) 𝑢(𝑡)  b. (0.5 + 𝑒 −𝑡 + 1.5𝑒 −2 ) 𝑢(𝑡)c. (1 + 2𝑒 −𝑡 − 2𝑒 −2𝑡 ) 𝑢(𝑡)d. (0.5 + 2𝑒 −𝑡 + 2𝑒 −2𝑡 ) 𝑢(𝑡)