Bangalore University

IV Semester B Sc(Hons.)Mathematics (Major)

(NEP-2021-2022 Onwards)
MATDSCT-4.1:Partial Differential Equations and Integral Transforms
Model Question Paper-I
1
Time: 2 2 Hrs. Max.Marks:60
PART-A
I. Answer any four of the following: 4 x 2=8
1. Solve: p2+q2=1
2. Solve p+q= sinx+siny
3. Solve (𝐷 − 𝐷’ − 1)(𝐷 + 𝐷’ − 2)𝑧 = 0
4. Find the Laplace transform of 𝑡 3 𝑒 2𝑡
𝑠2 −3𝑠+4
5. Find 𝐿−1 ( 𝑠3 )
6. Find 𝑎0 in the Fourier series of 𝑒 𝑥 in (-π,π )

PART-B
II. Answer any four of the following: 4 x 5=20
7. Form the partial differential equation by eliminating the arbitrary functions from
𝑧 = 𝑓(𝑥 − 𝑎𝑦) + 𝑔(𝑥 + 𝑎𝑦).
8. Solve x(y-z)p+y(z-x)q=z(x-y).
9. Solve (D2-2DD’+D’2)z=ex+2y.
𝑐𝑜𝑠𝑎𝑡−𝑐𝑜𝑠𝑏𝑡
10. Find L [ ]
𝑡
1
11. State Convolution Theorem. Using convolution theorem find 𝐿−1 (𝑠2 (𝑠+1))
12. Find the half range Fourier cosine series for the function f(x)=2x-1 in (0,2)

PART-C
III. Answer any four of the following: 4 x 8=32
13. Find the complete integral of z2(p2+q2+1)=1 by Charpit’s method.
14. Reduce the equation 𝑟 + 2𝑠 + 𝑡 = 0 to canonical form
𝜕2 𝑢 𝜕2 𝑢
15. Solve the equation = 𝑐 2 𝜕𝑥 2 subject to the conditions
𝜕𝑡 2

(i)u(0,t)=0, u(l,t)=0 for all t

𝜕𝑢
(ii) u(x,0)=k(lx –x2) and = 0 𝑎𝑡 𝑡 = 0
𝜕𝑡

16. Solve 𝑦 ′′ (𝑡) + 2𝑦 ′ (𝑡) + 𝑦(𝑡) = 𝑡𝑒 −𝑡 given that 𝑦(0) = 1 and 𝑦 ′ (0) = −2 using Laplace
transform.
17. Obtain the Fourier series for 𝑓(𝑥) = 𝑥 − 𝑥 2 in the interval [– 𝜋, 𝜋] and hence deduce that
1 1 𝜋2
1− + − ⋯ … … . . =
22 32 12
18. Define the finite Fourier cosine transform. And hence find the Fourier cosine Transform of
𝑒 −6𝑥 in the range 0 ≤ 𝑡 ≤ 1
 Bangalore University
IV Semester B Sc(Hons.)Mathematics (Major)
(NEP-2021-2022 Onwards)
MATDSCT-4.1:Partial Differential Equations and Integral Transforms
Model Question Paper-II
1
Time: 2 2 Hrs. Max.Marks:60
PART-A
I. Answer any four of the following: 4 x 2=8
1. Form the partial differential equation by eliminating the arbitrary constants from
z=(x+a)(y+b)
2. Solve p=eq.
3. Solve (2DD’+D’2 -3D’ )z=0
4. Find the Laplace transform of 𝑠𝑖𝑛5𝑡. 𝑐𝑜𝑠3𝑡
𝑠
5. Find 𝐿−1 ( )
(𝑠−3)3
6. Define the Finite Fourier cosine and sine transforms.

PART-B
II. Answer any four of the following: 4 x 5=20
𝜕𝑧 𝜕𝑧
7. Solve (mz-ny)𝜕𝑥 +(nx-lz) 𝜕𝑦 = ly-mx.

8. Solve z2(x2+p2+q2)=1 by taking the substitution u = logx.

9. Solve (D2-5DD’+4D’2)z = sin(4x+y)
𝑠2 +1
10. Find f(t) if L[f(t)] = log(𝑠(𝑠+1)_)
3𝑠+7
11. Find 𝐿−1 (𝑠2 +6𝑠+9)
12. Find the half range Fourier sine series for the function f(x)=1-x in (0,π)

PART-C
III. Answer any four of the following: 4 x 8=32
13. Find the complete integral of p(1+q2)+(b-z)q=0 by Charpit’s method
𝜕2 𝑧 𝜕2 𝑧
14. Reduce 𝜕𝑥 2 + 𝑥 2 𝜕𝑦 2 = 0 to canonical form
𝜕𝑢 𝜕2 𝑢
15. Solve the equation = 16 𝜕𝑥 2 subject to the condition
𝜕𝑡

(i)u(0,t)=0 and u(1,t)=0 for all t≥ 0

(ii) u(x,0)=x2 –x ,0≤ 𝑥 ≤ 1
16. Solve 𝑦 ′′ (𝑡) + 2𝑦 ′ (𝑡) + 5𝑦(𝑡) = 𝑒 −𝑡 𝑠𝑖𝑛𝑡 given that 𝑦(0) = 0 and 𝑦 ′ (0) = 1 using Laplace
transform.
𝑥 𝑖𝑓 0 < 𝑥 < 1
17. Obtain the Fourier series for 𝑓(𝑥) = {
2−𝑥 𝑖𝑓 1 < 𝑥 < 2
1 𝑖𝑓 |𝑥| < 𝑎
18. Define the Fourier Transform. And find the Fourier Transform of 𝑓(𝑥) = {
0 𝑖𝑓 |𝑥| > 𝑎
 Bangalore University
IV Semester B Sc(Hons.)Mathematics (Major)
(NEP-2021-2022 Onwards)
MATDSCT-4.1:Partial Differential Equations and Integral Transforms
Model Question Paper-III
1
Time: 2 2 Hrs. Max.Marks:60
PART-A
I. Answer any four of the following: 4 x 2=8
1. Solve: p2q3=1
2. Solve pey=qex
3. Find the particular integral of the equation (D2-2DD’+D’2)z=ex+2y
4. Find the Laplace transform of 𝑒−3𝑡 (3𝑠𝑖𝑛4𝑡 − 9𝑐𝑜𝑠4𝑡)
(1+2𝑠)2
5. Find 𝐿−1 ( 𝑠4 )
6. Find the half range Fourier sine series of the function f(x)=𝑒 𝑥 in the interval [0,π]

PART-B
II. Answer any four of the following: 4 x 5=20
7. Form the partial differential equation by eliminating the arbitrary functions from
𝑙𝑥 + 𝑚𝑦 + 𝑛𝑧 = 𝜑(𝑥 2 + 𝑦 2 + 𝑧 2 )
8. Solve (y-z)p+(z-x)q = x-y
9. Solve (D2-2DD’+D’2)z=12xy.
𝑠+8
10. Find 𝐿−1 (𝑠2 +4𝑠+5)

11. State Convolution Theorem. Verify convolution theorem for 𝑓(𝑡) = 𝑒 𝑡 𝑎𝑛𝑑 𝑔(𝑡) = 𝑐𝑜𝑠𝑡
12. Find the half range Fourier cosine series for the function f(x)=(x-1)2 in the interval (0,1)

PART-C
III. Answer any four of the following: 4 x 8=32
13. Find the complete integral of (p2+q2)y=qz by Charpit’s method.
𝜕2 𝑧 𝜕2 𝑧
14. Reduce 𝜕𝑥 2 = 𝑥 2 𝜕𝑦 2 to canonical form

u  2u
15. Solve the equation = 4 2 subject to the condition
t t
(i)u(0,t)=0 ,u(1,t)=0 for all t≥ 0
(ii) u(x,0)= x- x2 ,0≤ 𝑥 ≤ 1.
16. Solve 𝑥 ′′ (𝑡) + 4𝑥′(𝑡) + 4𝑥(𝑡) = 4𝑒 −2𝑡 given that 𝑥(0) = −1 and 𝑥 ′ (0) = 4 using Laplace
transform.
17. Obtain Fourier series of the function 𝑓(𝑥) = 𝑥 + 𝑥 2 in [– 𝜋, 𝜋]
1 − 𝑥 2 𝑖𝑓 |𝑥| ≤ 1
18. Define the Fourier Transform. Find the Fourier Transform of 𝑓(𝑥) = {
0 𝑖𝑓 |𝑥| > 1