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

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

Solve the following initial value problem:𝑦=∫(3𝑥2−6)d𝑥,𝑦(1)=2

Use Lagrange method to write the general solution of the following partial differentialequations.

Solve the system of equations using the substitution method. State your final answer as an ordered pair. DO NOT include spaces in your answer.y=6x𝑦=6𝑥2x−5y=02𝑥−5𝑦=0Solution:

Find the value of variables :1𝑞+2𝑚−10=0 and 6𝑞+2𝑚−20=0𝑞=3 ; 𝑚=3𝑞=3 ; 𝑚=5𝑞=2 ; 𝑚=4𝑞=1 ; 𝑚=