Ordinary Differential Equations Questions and Answers – Special Functions – 2 (Beta)

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This set of Ordinary Differential Equations Question Paper focuses on “Special Functions -2 (Beta)”.

1. β(m, n) = β(n, m). Is the statement true?
a) True
b) False
View Answer

Answer: a
Explanation: L.H.S. = \(\beta(m, n) = \frac{\Gamma(m).\Gamma(n)}{\Gamma(m+n)}. \)
R.H.S. \( = \beta(n, m) = \frac{\Gamma(n).\Gamma(m)}{\Gamma(n+m)} = \frac{\Gamma(m).\Gamma(n)}{\Gamma(m+n)} = \) L.H.S.
Therefore, \(\beta(m, n) = \beta(n, m).\)
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2. Which of the following function is not called the Euler’s integral of the first kind?
a) \(\beta(m, n) = \int_0^1 x^{m-1} (1-x)^{n-1} dx (m>0, n>0) \)
b) \(\beta(m, n) = \int_0^{π/2} (sin⁡θ)^{2m-1} (cos⁡θ)^{2n-1} dθ \)
c) \(\beta(m, n) = \int_0^∞ \frac{y^{n+1}}{(1+y)^{m+n}} dy \)
d) \(\beta(m, n) = 2 \int_0^{π/2} (sinθ)^{2m-1} (cos⁡θ)^{2n-1} dθ \)
View Answer

Answer: b
Explanation: Euler’s integral of the first kind is nothing but Beta function. So, here only \( \beta(m, n) = \int_0^{π/2}(sin⁡θ)^{2m-1} (cos⁡θ)^{2n-1} dθ\) is not the definition of Beta function.

3. Which of the following is not the definition of Beta function?
a) \(\beta(m, n) = 2\int_0^1 x^{m-1} (1-x)^{n-1} dx (m>0, n>0) \)
b) \(\beta(m, n) = 2\int_0^{π/2}(sin⁡θ)^{2m-1} (cos⁡θ)^{2n-1} dθ \)
c) \(\beta(m, n) = \int_0^∞ \frac{y^{n+1}}{(1+y)^{m+n}} dy \)
d) \(\beta(m, n) = \int_0^1 \frac{x^{m-1}+x^{n-1}}{(1+x)^{m+n}} dx \)
View Answer

Answer: a
Explanation: \(\beta(m, n)\) can be written as either \(\int_0^1 x^{m-1} (1-x)^{n-1} dx \, (m>0, n>0)\) (or)
\(= 2 \int_0^{π/2}(sin⁡θ)^{2m-1} (cos⁡θ)^{2n-1} dθ \)
(or)
\( = \int_0^∞ \frac{y^{n+1}}{\left(1+y\right)^{m+n}} dy \;or \int_0^1 \frac{x^{m-1}+x^{n-1}}{\left(1+x\right)^{m+n}} dx\).
So the correct answer is \(2\int_0^1 x^{m-1} (1-x)^{n-1} dx (m>0, n>0) \) which is actually not the formula for Beta function.

4. What is the value of \(\beta(m, \frac{1}{2}) \)?
a) β(m, m)
b) 22m-1 β(m, m)
c) 22m+1 β(m, m)
d) 22m β(m, m)
View Answer

Answer: b
Explanation: \(\beta(m, \frac{1}{2}) = 2\int_0^{π/2}(sin⁡θ)^{2m-1} dθ \)
\(\beta(m, m) = 2 \int_0^{π/2}(sin⁡θ)^{2m-1} (cos⁡θ)^{2m-1} dθ \)
\( = 2^{-2m+2} \int_0^{π/2}(2 sinθ cos⁡θ)^{2m-1} dθ \)
Substituting 2θ=φ,
\( = 2^{-2m+1} \int_0^π sin⁡φdφ \)
\( = 2^{-2m+1}.2.\int_0^{π/2}sin⁡φdφ \)
\( = \frac{1}{2^{2m-1}} \beta(m, \frac{1}{2}). \)

5. What is the value of β(3,2)?
a) \(\frac{1}{14} \)
b) \(\frac{1}{16} \)
c) \(\frac{1}{12} \)
d) \(\frac{1}{10} \)
View Answer

Answer: c
Explanation: \(\beta(3, 2) = \frac{\Gamma(3).\Gamma(2)}{\Gamma(3+2)} \)
\( = \frac{2!1!}{4!} = \frac{1}{12}.\)
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6. What is the value of \(\beta(\frac{1}{4},\frac{3}{4})\)?
a) \(\pi \)
b) \(\sqrt{2}\pi \)
c) \(\sqrt{2\pi} \)
d) \({2}\pi \)
View Answer

Answer: b
Explanation: \(\beta(\frac{1}{4}, \frac{3}{4}) = \frac{\Gamma(\frac{1}{4}).\Gamma(\frac{3}{4})}{\Gamma(\frac{1}{4}+\frac{3}{4})} \)
\( = \frac{\pi}{sin⁡ \frac{π}{4}} = \sqrt{2}\pi. \)

7. What is the value of \(\int_0^{π/2}\sqrt{sin⁡θdθ} + \int_0^{π/2}\sqrt{cos⁡θ⁡dθ} \)?
a) \(8\sqrt{\pi} \frac{\Gamma(\frac{3}{4})}{\Gamma(\frac{1}{4})} \)
b) \(4\sqrt{\pi} \frac{\Gamma(\frac{3}{4})}{\Gamma(\frac{1}{4})} \)
c) \(8\sqrt{\pi} \frac{\Gamma(\frac{1}{4})}{\Gamma(\frac{3}{4})} \)
d) \(4\sqrt{\pi} \frac{\Gamma(\frac{1}{4})}{\Gamma(\frac{3}{4})} \)
View Answer

Answer: a
Explanation: \(\int_0^{π/2}\sqrt{sin⁡θdθ} + \int_0^{π/2}\sqrt{cos⁡θ⁡dθ} \)
\( = \beta(\frac{3}{4}, \frac{1}{2}) + \beta(\frac{3}{4}, \frac{1}{2}) \)
\( = 2 \beta(\frac{3}{4}, \frac{1}{2}) \)
\( = 2\frac{\Gamma(\frac{1}{2}).\Gamma(\frac{3}{4})}{\Gamma(\frac{1}{2}+\frac{3}{4})} \)
\( = 2\sqrt{\pi} \frac{\Gamma(\frac{3}{4})}{\Gamma(\frac{1}{4})} . \frac{1}{(\frac{1}{4})} \)
\( = 8\sqrt{\pi} \frac{\Gamma(\frac{3}{4})}{\Gamma(\frac{1}{4})}. \)

8. What is the value of \(\int_0^∞ \frac{1}{(1+y)^5} dy \)?
a) 12
b) 13
c) 14
d) 15
View Answer

Answer: c
Explanation: \(\int_0^∞ \frac{y^{1-1}}{(1+y)^5} dy \)
n=1 and m + n = 5 which implies m=4.
\(\beta(4, 1) = \frac{\Gamma(4).\Gamma(1)}{\Gamma(4+1)} \)
\(\frac{3!0!}{4!} = \frac{1}{4} \).

9. What is the value of \(\int_0^1 \frac{dx}{\sqrt{1+x^4}}\)?
a) \(\beta(\frac{1}{4}, \frac{1}{2}) \)
b) \(\frac{1}{4\sqrt{2}}\beta(\frac{1}{4}, \frac{1}{2}) \)
c) \(\frac{1}{3\sqrt{2}}\beta(\frac{1}{3}, \frac{1}{2}) \)
d) \(\frac{1}{4\sqrt{3}}\beta(\frac{1}{4}, \frac{1}{3}) \)
View Answer

Answer: b
Explanation: Substitute x2 = tan⁡θ
Therefore θ varies from 0 to \(\frac{\pi}{4}.\)
\(\int_0^{π/4} \frac{(secθ)^2}{2 sec⁡θ \sqrt{tan⁡θ}} dθ \)
\(= \int_0^{π/4}\frac{dθ}{2\sqrt{sin⁡θ cos⁡θ}} \)
\(= \frac{1}{4\sqrt{2}} \beta(\frac{1}{4}, \frac{1}{2}).\)
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10. What is the value of \(\int_0^1\frac{(y^5+y^2)}{(1+y)^9} dy \)?
a) \(\frac{1}{158} \)
b) \(\frac{2}{167} \)
c) \(\frac{1}{146} \)
d) \(\frac{1}{168} \)
View Answer

Answer: d
Explanation: m-1 = 5 => m=6 and n-1 = 2 => n=3.
\(\beta(6, 3) = \frac{\Gamma(6).\Gamma(3)}{\Gamma(6+3)} \)
\(= \frac{5!2!}{8!} = \frac{1}{168}. \)

11. Solve using the Beta function. \(\int_0^1 x^{-2} (1-x)^{-3} dx. \)
a) Can be solved using a Beta function with m = -1 and n = -2
b) Can be solved using a Beta function with m = 1 and n = -2
c) Can be solved using a Beta function with m = -1 and n = 2
d) Can’t be solved using the Beta function
View Answer

Answer: d
Explanation: This function can’t be solved using the Beta function as m and n have negative values. Beta function can’t be solved if m and n are negative numbers.

12. What is the value of \(\int_0^∞ (sech x)^5 dx \)?
a) \(\frac{3\pi}{80} \)
b) \(\frac{3\pi}{240} \)
c) \(\frac{3\pi}{16} \)
d) \(\frac{\pi}{240} \)
View Answer

Answer: c
Explanation: \(\int_0^∞ (sech x)^5 dx = \frac{2^5}{4} \beta(\frac{5}{2}, \frac{5}{2}) \)
\(\displaystyle = \frac{8 \Gamma(\frac{5}{2})\Gamma(\frac{5}{2})}{\Gamma(5)} = \frac{(8*\frac{3}{2}*\frac{1}{2}*\sqrt{\pi}*\frac{3}{2}*\frac{1}{2}*\sqrt{\pi})}{24} = \frac{9π}{48}=\frac{3π}{16}. \)

13. What is the value of \(\beta(\frac{9}{2},3) \)?
a) \(\frac{16}{1287} \)
b) \(\frac{16}{1278} \)
c) \(\frac{14}{1287} \)
d) \(\frac{16}{127} \)
View Answer

Answer: a
Explanation: \(\beta(\frac{9}{2},3) = \frac{\Gamma(\frac{9}{2})\Gamma(3)}{\Gamma(\frac{9}{2}+3)} \)
\( = \frac{\Gamma(\frac{9}{2})2}{\frac{13}{2}*\frac{11}{2}*\frac{9}{2}*\Gamma(\frac{9}{2})} = \frac{16}{1287}.\)
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14. What is the value of \(\int_0^1 x^5 (1-x)^6 dx\)?
a) \(\frac{1}{12*11*10*9*8*7} \)
b) \(\frac{1}{12*11*10*9*8} \)
c) \(\frac{1}{12*11*10*9*8*7*6} \)
d) \(\frac{1}{12*11*10*9*8*7*6*5} \)
View Answer

Answer: c
Explanation: Here, m-1 = 5 which implies m=6 and n-1 = 6 which implies n=7.
\(\beta(6,7)= \frac{\Gamma(6)\Gamma(7)}{\Gamma(13)} = \frac{1}{12*11*10*9*8*7*6}. \)

Sanfoundry Global Education & Learning Series – Ordinary Differential Equations.

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Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He is Linux Kernel Developer & SAN Architect and is passionate about competency developments in these areas. He lives in Bangalore and delivers focused training sessions to IT professionals in Linux Kernel, Linux Debugging, Linux Device Drivers, Linux Networking, Linux Storage, Advanced C Programming, SAN Storage Technologies, SCSI Internals & Storage Protocols such as iSCSI & Fiber Channel. Stay connected with him @ LinkedIn