EndSem Jan2021
EndSem Jan2021
EndSem Jan2021
Differential Equations & Complex END SEM (Session- Jan 2021) Date–05/07/2021
Analysis (MAIR 21) (1st year B.Tech)
1. Complex analysis:
a) Check whether the function 𝑓(𝑧) = |𝑧|2 is analytic or not? [1]
b) Let 𝑧 = 𝑥 + 𝑖 𝑦 and 𝑓(𝑧) = 𝑢(𝑥, 𝑦) + 𝑖 𝑣(𝑥, 𝑦) is an analytic function with [2]
𝑢 − 𝑣 = 𝑒 −𝑥 [(𝑥 − 𝑦) sin 𝑦 − (𝑥 + 𝑦) cos 𝑦].
Then find 𝑢(𝑥, 𝑦) and 𝑣(𝑥, 𝑦).
c) If 0 < 𝑟 < 𝑅, and 𝐶: |𝑧| = 𝑟 then evaluate the contour integral [2]
𝑅+𝑧
𝐼=∮ 𝑑𝑧.
𝑧(𝑅 − 𝑧)
𝐶
2. Use the concept of Laplace transform to find the solution of the following;
a) Find [2]
−1
(𝑠 + 1)𝑒 −𝜋𝑠
𝐿 [ 2 ].
𝑠 +𝑠+1
b) Let Γ[𝑛] ∶= (𝑛 − 1)!, and [3]
1
d) Solve [3]
𝑑3 𝑦 𝑑2𝑦 𝑑𝑦 𝜋
3
− 2 2
+5 = 0, 𝑦(0) = 0, 𝑦 ′ (0) = 1, 𝑦 ( ) = 1.
𝑑𝑡 𝑑𝑡 𝑑𝑡 8
3. Find the complete solution of the following ordinary differential equations.
a) [2]
𝑑4 𝑦 𝑑3𝑦 𝑑2𝑦 𝑑𝑦
𝑥4 4
+ 6𝑥 3
3
+ 4𝑥 2
2
− 2𝑥 − 4𝑦 = 𝑥 2 + 2 cos(log 𝑥)
𝑑𝑥 𝑑𝑥 𝑑𝑥 𝑑𝑥
b) [2]
𝑑2𝑦 𝑑𝑦 1
(𝑥 + 𝑎)2 + 3(𝑥 + 𝑎) + 𝑦 = ,
𝑑𝑥 2 𝑑𝑥 (1 − 𝑥)2
here 𝑎 is a constant.
4. Find the complete integrals of the following partial differential equations.
a) 𝑥(𝑦 2 + 𝑧)𝑝 − 𝑦(𝑥 2 + 𝑧)𝑞 = 𝑧(𝑥 2 + 𝑦 2 ) [2]
b) 𝑥(𝑥 2 + 3𝑦 2 )𝑝 − 𝑦(3𝑥 2 + 𝑦 2 )𝑞 = 2𝑧(𝑦 2 − 𝑥 2 ) [2]
c) cos(𝑥 + 𝑦) 𝑝 + sin(𝑥 + 𝑦) 𝑞 = 𝑧 [2]
d) 2𝑥𝑧 − 𝑝𝑥 2 − 2𝑞𝑥𝑦 + 𝑝𝑞 = 0 [2]
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