Higher Engineering Mathematics

462 DIFFERENTIAL EQUATIONS For line 5,wherex 0 = 1 .8: y 1 =y 0 +h(y′) 0 = 5. 312 +(0.2)(2.488)=5.8096 and (y′) 0 =3(1+ 1 .8)− 5 ...

NUMERICAL METHODS FOR FIRST ORDER DIFFERENTIAL EQUATIONS 463 I A graph of the solution of dy dx +y= 2 x, with initial conditions ...

464 DIFFERENTIAL EQUATIONS 0 0.1 0.2 0.3 0.4 0.5 x 2.5 3.0 y 2.0 Figure 49.8 Euler’s method of numerical solution of differentia ...

NUMERICAL METHODS FOR FIRST ORDER DIFFERENTIAL EQUATIONS 465 I Table 49.7 xy 2.0 1 2.2 1.2 2.4 1.421818 2.6 1.664849 2.8 1.92871 ...

466 DIFFERENTIAL EQUATIONS Table 49.8 xy y′ 0 2 2 0.1 2.205 2.105 0.2 2.421025 2.221025 0.3 2.649232625 2.349232625 0.4 2.89090 ...

NUMERICAL METHODS FOR FIRST ORDER DIFFERENTIAL EQUATIONS 467 I % error with Euler-Cauchy method = ( 2. 649858808 − 2. 649232625 ...

468 DIFFERENTIAL EQUATIONS For line 6,x 1 = 2. 0 yP 1 =y 0 +h(y′) 0 = 5. 85212176 + 0 .2(2.54787824) = 6. 361697408 yC 1 =y 0 +^ ...

NUMERICAL METHODS FOR FIRST ORDER DIFFERENTIAL EQUATIONS 469 I Obtain a numerical solution of the differential equation 1 x dy ...

470 DIFFERENTIAL EQUATIONS 6.yn+ 1 =yn+ h 6 {k 1 + 2 k 2 + 2 k 3 +k 4 }and when n=0: y 1 =y 0 + h 6 {k 1 + 2 k 2 + 2 k 3 +k 4 } ...

NUMERICAL METHODS FOR FIRST ORDER DIFFERENTIAL EQUATIONS 471 I Table 49.16 Euler’s Euler-Cauchy Runge-Kutta method method method ...

472 DIFFERENTIAL EQUATIONS 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 2.181267 2.2631 ...

NUMERICAL METHODS FOR FIRST ORDER DIFFERENTIAL EQUATIONS 473 I Table 49.18 Euler’s Euler-Cauchy Runge-Kutta method method method ...

Assign-13-H8152.tex 23/6/2006 15: 13 Page 474 Differential equations Assignment 13 This assignment covers the material contained ...

I Differential equations 50 Second order differential equations of the form a d 2 y dx 2 +b dy dx +cy= 0 50.1 Introduction An eq ...

476 DIFFERENTIAL EQUATIONS (c) If the roots of the auxiliary equation are: (i)real and different, saym=αandm=β, then the general ...

SECOND ORDER DIFFERENTIAL EQUATIONS (HOMOGENEOUS) 477 I Whent=0, dy dt = 3 thus 3=(0+B) 4 3 e^0 +Ae^0 i.e. 3= 4 3 B+Afrom which, ...

478 DIFFERENTIAL EQUATIONS 6 d^2 y dx^2 + 5 dy dx − 6 y=0; whenx=0,y=5 and dy dx =−1. [ y=3e 2 3 x+2e− 3 2 x ] 4 d^2 y dt^2 ...

SECOND ORDER DIFFERENTIAL EQUATIONS (HOMOGENEOUS) 479 I When t=0, dV dt = 3 ω, thus 3 ω=Aω−Bω, i.e. 3 =A−B (2) From equations (1 ...

480 DIFFERENTIAL EQUATIONS Thus 4=A+B (1) Velocity, dx dt =− 2 Ae−^2 t− 4 Be−^4 t dx dt =8 cm/s whent=0, thus 8 =− 2 A− 4 B (2) ...

I Differential equations 51 Second order differential equations of the form a d 2 y dx^2 + b dy dx + cy = f(x) 51.1 Complementar ...