Higher Engineering Mathematics

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Differential equations I 46 Solution of first order differential equations by separation of variables 46.1 Family of curves Inte ...

444 DIFFERENTIAL EQUATIONS Now try the following exercise. Exercise 177 Further problems on families of curves Sketch a family ...

SOLUTION OF FIRST ORDER DIFFERENTIAL EQUATIONS BY SEPARATION OF VARIABLES 445 I Since 5 dy dx + 2 x=3 then dy dx = 3 − 2 x 5 = 3 ...

446 DIFFERENTIAL EQUATIONS Find the equation of the curve if it passes through the point ( 1,^13 ) . [ y= 3 2 x^2 − x^3 6 − 1 ] ...

SOLUTION OF FIRST ORDER DIFFERENTIAL EQUATIONS BY SEPARATION OF VARIABLES 447 I αis the temperature coefficient of resistance of ...

448 DIFFERENTIAL EQUATIONS The rate of cooling of a body is given by dθ dt =kθ, wherekis a constant. Ifθ= 60 ◦C when t=2 minut ...

SOLUTION OF FIRST ORDER DIFFERENTIAL EQUATIONS BY SEPARATION OF VARIABLES 449 I Integrating both sides gives: 1 2 ln (1+x (^2) ) ...

450 DIFFERENTIAL EQUATIONS Problem 13. For an adiabatic expansion of agas Cv dp p +Cp dV V =0, whereCpandCvare constants. Givenn ...

I Differential equations 47 Homogeneous first order differential equations 47.1 Introduction Certain first order differential eq ...

452 DIFFERENTIAL EQUATIONS Thus, the particular solution is: y x =−lnx+ 2 ory=−x(lnx−2)ory=x( 2 −lnx) Problem 2. Find the partic ...

HOMOGENEOUS FIRST ORDER DIFFERENTIAL EQUATIONS 453 I (iii) Substituting foryand dy dx gives: v+x dv dx = 2 x^2 + 12 x(vx)− 10 (v ...

454 DIFFERENTIAL EQUATIONS Wheny=3,x=1, thus: ln ( 9 1 − 1 ) =ln 1+c from which,c=ln 8 Hence, the particular solution is: ln ( y ...

I Differential equations 48 Linear first order differential equations 48.1 Introduction An equation of the form dy dx +Py=Q, whe ...

456 DIFFERENTIAL EQUATIONS equation. Given boundary conditions, the par- ticular solution may be determined. 48.3 Worked problem ...

LINEAR FIRST ORDER DIFFERENTIAL EQUATIONS 457 I Hence from equation (4):yex=xex−ex+c, which is the general solution. Whenx=0,y=2 ...

458 DIFFERENTIAL EQUATIONS Whenx=−1, − 6 =− 3 A, from which,A= 2 Whenx=2, 3 = 3 B, from which,B= 1 Hence ∫ 3 x− 3 (x+1)(x−2) dx ...

LINEAR FIRST ORDER DIFFERENTIAL EQUATIONS 459 I The concentration,C, of impurities of an oil purifier varies with timetand is d ...

Differential equations 49 Numerical methods for first order differential equations 49.1 Introduction Not all first order differe ...

NUMERICAL METHODS FOR FIRST ORDER DIFFERENTIAL EQUATIONS 461 I The approximation used with Euler’s method is to take only the fi ...