JVOTHY INSTITUTE OF TECHNOLOGY JATHEMA’ ‘SWMECK:. ASSIGNMENT 4 wy |. a. Using the Taylor's series method, solve the inital value problem “7 = x"y—1, (0)-1 atthe point x=0.1. (6 M)(DEC-12,08,JUN-08,11;DEC-15) », Employ the Taylor's series method to find an approximate solution to find y at x = 0.1 given dy ~y*,y(0)=1 by considering upto 4" degree term. (6M) (DEC-14) dy 2. a, Solve 2 = 2y+ 3c", y(0)=0 using Taylor's series method and find y(0.1). Compare the numerical ix solution with the analytical solution, (6 M)JUN-09,DEC-2011, 12, 13,16) b, Solve the following by Euler's Modified method y* = log(x-+y),y(0)=2 to find y(0.4) by taking h = 0.2-Perform two modifications at each step. - (CIM)DEC-10) 2y?,y(0)=0. Find y(0.25) using the modified Euler's method. Perform two modifications. (7M\DEC-11) y(0)=1 at the points x=0.2. Take h 'mploy the fourth order Runge-Kutta method to solve yy (7M)(DEC-12,JUN-08,11) : «ds utta method of fourth order to solve + for y(0.1) given that y=1, when x=0. 4. a. Apply Runge-K: de (7M)IAN-17) b, Using fourth order Runge-Kutta method to sols (0) = Latthe point x = 0.1 (7M)(DEC-15) te (0.4) by Milne’s method. (7M)(JUN-09) x+y", using Milne's predictor-corrector method find y(1.4).AApply two corrections, (7M)(JAN-17) 1979" top S(IP1O, ST) 1.233, (1.2)°1 S48 and y(1.3)=1.979, compute y(1-4) by Adams-Bashforth a, Given that y!=x"(L ty) method. . (MYIUN-10,11,DEC-09) b, Given that y!=1/2 xy and y(O)> 1, y(0.1)° 1.0025, (0.2) 10101, y(0.3)=1.0228, compute y(0.4) by Adams-Bashforth method, ON) (DEC-13, JUN-14) PTO ‘a, Using the Runge-Kutta method, find the solution at x=0.1 of the differential equation “2 Patx=02wath ¥(0)=1and 2(0)=0 take h =0.2 (CIM)QULA 13, JUNTA TUN-TS) . Using the Runge-Kutta method, find the solution at x0. of the differential equation ay ia ; : a. x a — 2xy = Lunder the conditions y(O)*1 y'(0y0. Take A= 0.1 OM\DEC-09,12) we 8. a. Using the Milne’s method, obtain an approximate solution at the point x=0.4 of the problem os +3x * ~tyent YCO=1, y"(O)O.1, Given that y(O.1)=1,03995, y(0.2)=1.138036, y(0.3)"1.29865, y'(0.1)0.6955, y'(0.29=1.258, y'@.3F1.873, COM\DEC-12,10), . . ' th b. Using the Milne’s method, obtain an approximate solution at the point x=0.8 of “—% dy * =2y and y & y' are giver a gg TA COV, y* (=I, y(0.2)=0.2027, y(0.4)=0.4228, y(0.6)=0.6841, y(0.2 1.041,0.6955, y'(0.4)=1.179, y"(0.6)= 1.465 (7M) (DEC- 11 JAN-16)JYOTHY INSTITUTE OF TECHNOLOGY DEPARTMENT OF MATHEMATICS ‘Transform Calculus, Fourier Series and Numerical Techniques-18MAT31 Assignment -02 , ara 1a, Derive Buler’s equation for extremal value in the form 35— ay (gyi b, Solve the va jon problem 5 {2(x-+y + y')dx = 0, under the conditions y(0) = 1,y(1) Find the geodesics on a surface given that the arc length onthe surface is S = f2? |x(1 + y")dx 2.0. A heavy cable hangs fieely’under gravity beiween twafixed points. Show that the shape of the cable is a catenary. (Hanging cable problem) b. Find the path in which a particle in the absence of friction will slide from one point to another in the shortest time under the action of gravity. (Brachistochrone problem) 3 a. Find () L{SineSin2xSin3e} Gn Le Cosh} L b.Find DL{r (Sin? r Cos? 1)} cy LIPe™ Sin24 } city { 4. a Find the value of [re cos2 dt using Laplace Transform 0 eo Sint ; . Find the Laplace Transform of and hence deduce that f Q 5 .a, Find the Laplace Transform of the periodic function with period om o io-| ESinot, O2n HAsO} 1 if O2 v, Ises2 on . it elt S(t) an nee F(t) in terms of Unit Step function and hence find its LT ; _ a, Find (i784) — my Ete 7. a, Find (iL {os i me mare] 7 { __ Ss+3 wf uy Tat b. Find (i) ® eo a wo Fea Ba ros | b, Find £ { we 2ye { using Convolution Theorem } using Convolution Theorem © 0 ig 2) = 9") 50) = 6 using LT 9.a. Solve * i with 1002090) yog 2) 6 sing ot b. Solve "+6" #9" 12% Submission Date: 22-10-2019Jyothy Institute of Technology DEPARTMENT OF MATHEMATICS Subjects Transform Calculus, Fourler Seles and Nomerient Techniques (Common to all busnch ASSIGNMENTAN 1a. Find a Fourier series to represent (x) = x? fom x= rr Co = amd deduce that, > 1nd the Fourier series for the fanetion f(x) © x(n x) over (0,2) and deduce that, 5 maha. (79H (WEC-2010,s0N-2016) 2.4, Expand th ineton f(x) = x sina asa Fourier sevies i the interval [n,n Dede hat (7M) (JUN-2008, 2015) me and deduce = 3. Obtain the Fourier series forthe finction fe) = ("0 _ gy SERED (7M) (JUN-2098, 2013, DEC-2011) ». Find Fourier Cosine series of (x)= sin() x, where m isa postive integer. (6 M\(JUN-2014) 4.a. Obtain half range cosine series forthe function ke for O0 (7M) (Jun-2015) 1, |xl0. (6 M) (Dee-2.009) ind the inverse Fourier sine transform of +e" ,a>0 (7M) (Dec- 2011) 4a. Find the Z- transform of : i) (2n-1)? ii) eof 2 4) iti) 3n Asin Eso (7M) (Dee-2010) b. Find the Z-transforms of : i) Cosh os +0) ii)sin@n¥5) (7 M) (Jun-2009) ¢. Find the Z-transform of i) a" sind ii) a” e!” jii)a) coshn0 and sinhnd ind -2zcosO+1 z-cos0) 2zcosO +1 (7M) (Sun-201 0) ii) Z(sinn0)= — 5a. Prove that : i) Z(cosn0) = 3 1b. Find the 2. transform of, [E] =e% hence find Zp bealoez, etsl ©. Find Z(e* sinno) and Zncosnd). 6 a. Obtain the inverse 427 Zetransform of (i) ——22-— 22 — @ Dagirer4 (7M) (Dec-2011, 201106) ee 228432 (2-29(@-4) (7M) (Jun-2011) rere (7M) (Jun-08) Gi) (7M) (Jun-2013) , 8: Gna) * with yo= yy 0 using Z-transforms. (7 M) (Jun -2011,09 , b. (i) Solve the difference equation y,.: + OY. + 9Y Dec- 08,13) = 2, with tp=3 , uy= 7 using z-transforms. (7M) (Dec-2010) by (Solve the difference equation u,,, ~ St.) + 64, = Land u,=1 for n=0,1,2,.. +6y, =u, with yo=0.y1 . (6M) ( Dee-2010) (ii)Find the resyfonse of thesystem ¥,.3 ~ Z-transform method. = 3y, = 3° +2 with yo = yy = 0. (7M) (Jun-2018) (iv)Obtain the solution of the difference equation J,» ~