Higher Engineering Mathematics, Sixth Edition
462 Higher Engineering Mathematics which is a statement calledTaylor’s series. Ifhis the interval between two new ordinatesy 0 a ...
Numerical methods for first order differential equations 463 This step by step Euler’s method can be continued and it is easiest ...
464 Higher Engineering Mathematics For line 2,wherex 0 = 0 .2andh= 0 .2: y 1 =y 0 +h(y′), from equation( 2 ) = 1 +( 0. 2 )(− 1 ) ...
Numerical methods for first order differential equations 465 For line 4,wherex 0 = 0 .3: y 1 =y 0 +h(y′) 0 = 2. 41 +( 0. 1 )( 2. ...
466 Higher Engineering Mathematics Table 49.5 x y 0 1 0.2 1 0.4 0.96 0.6 0.8864 0.8 0.793664 1.0 0.699692 (b) If the solution of ...
Numerical methods for first order differential equations 467 corrected value,yC 1 in the improved Euler method is given by: yC 1 ...
468 Higher Engineering Mathematics yC 1 =y 0 +^12 h[(y′) 0 +f(x 1 ,yP 1 )] = 2. 89090205 +^12 ( 0. 1 )[2. 49090205 +( 3. 1399922 ...
Numerical methods for first order differential equations 469 yP 1 =y 0 +h(y′) 0 = 4 + 0. 2 ( 2 )= 4. 4 yC 1 =y 0 +^12 h[(y′) 0 + ...
470 Higher Engineering Mathematics From Table 49.11 of Problem 5, by the Euler-Cauchy method, whenx= 1 .6,y= 5. 351368 % error i ...
Numerical methods for first order differential equations 471 49.5 The Runge-Kutta method TheRunge-Kuttamethodforsolvingfirst ord ...
472 Higher Engineering Mathematics Table 49.15 n xn k 1 k 2 k 3 k 4 yn 0 0 2 1 0.1 2.0 2.05 2.0525 2.10525 2.205171 2 0.2 2.1051 ...
Numerical methods for first order differential equations 473 Table 49.16 Euler’s Euler-Cauchy Runge-Kutta method method method E ...
474 Higher Engineering Mathematics Table 49.17 n xn k 1 k 2 k 3 k 4 yn 0 1.0 4.0 1 1.2 2.0 2.1 2.09 2.182 4.418733 2 1.4 2.18126 ...
Numerical methods for first order differential equations 475 The percentage error in the Runge-Kutta method when, say,x= 1 .6is: ...
Revision Test 14 This Revision Test covers the material contained in Chapters 46 to 49.The marks for each question are shown in ...
Chapter 50 Second order differential equations of the form a d 2 y dx 2 + b dy dx +cy= 0 50.1 Introduction An equation of the fo ...
478 Higher Engineering Mathematics 50.2 Procedure to solve differential equations of the form a d^2 y dx^2 +b dy dx +cy= 0 (a) R ...
Second order differential equations of the forma d^2 y dx^2 +b dy dx+cy=^0479 (d) Whent=0,y=3 hence 3=( 0 +B)e^0 ,i.e.B=3. Since ...
480 Higher Engineering Mathematics 3. d^2 y dx^2 + 2 dy dx + 5 y= 0 [y=e−x(Acos 2x+Bsin2x)] InProblems4to9,findtheparticularsolu ...
Second order differential equations of the forma d^2 y dx^2 +b dy dx+cy=^0481 (c) Since the roots are real and different,the gen ...
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