Higher Engineering Mathematics, Sixth Edition
482 Higher Engineering Mathematics Using the procedure of Section 50.2: (a) d^2 x dt^2 + 6 dx dt + 8 x=0 in D-operator form is ( ...
Chapter 51 Second order differential equations of the form a d 2 y dx 2 + b dy dx + cy = f (x) 51.1 Complementary function and p ...
484 Higher Engineering Mathematics Table 51.1Form of particular integral for different functions Type Straightforward cases Try ...
Second order differential equations of the forma d^2 y dx^2 +b dy dx+cy=f(x)^485 (ii) Substitutingmfor D gives the auxiliary equ ...
486 Higher Engineering Mathematics Now try the following exercise Exercise 189 Further problems on differential equations of the ...
Second order differential equations of the forma d^2 y dx^2 +b dy dx+cy=f(x)^487 −^23 =( 0 +B)e^0 +^13 e^0 , from which,B=−1. dy ...
488 Higher Engineering Mathematics from which, 2ke^2 x=3e^2 xandk=^32 Hence the P.I.,v=kx^2 e^2 x=^32 x^2 e^2 x. (vi) The genera ...
Second order differential equations of the forma d^2 y dx^2 +b dy dx+cy=f(x)^489 Equating coefficients of cos2xgives: 6 A− 13 B= ...
490 Higher Engineering Mathematics Now try the following exercise Exercise 191 Further problems on differential equations of the ...
Second order differential equations of the forma d^2 y dx^2 +b dy dx+cy=f(x)^491 (v) Substitutingvinto (D^2 +D− 6 )v= 12 x−50sin ...
492 Higher Engineering Mathematics Equating coefficients of exsin2xgives: − 3 C− 4 D− 2 C+ 4 D+ 2 C= 0 i.e.− 3 C=0, from which,C ...
Chapter 52 Power series methods of solving ordinary differential equations 52.1 Introduction Second order ordinary differential ...
494 Higher Engineering Mathematics In general, y(n)=ansin ( ax+ nπ 2 ) (2) For example, if y=sin 3x,then d^5 y dx^5 =y(^5 ) = 35 ...
Power series methods of solving ordinary differential equations 495 Note that ify=lnx,y′= 1 x ; if in equation (7), n=1theny′=(− ...
496 Higher Engineering Mathematics Thus, wheny=x^2 e^3 x,v=x^2 , since its third derivative is zero, andu=e^3 xsince thenth deri ...
Power series methods of solving ordinary differential equations 497 sin(x+ 2 π)≡sinx,sin ( x+ 3 π 2 ) ≡−cosx, and sin(x+π)≡−sinx ...
498 Higher Engineering Mathematics This equation is called arecurrence relation orrecurrence formula, because each recurring ter ...
Power series methods of solving ordinary differential equations 499 equation (13), each term is differentiated n times, which gi ...
500 Higher Engineering Mathematics Show that the power series solution of the dif- ferential equation:(x+ 1 ) d^2 y dx^2 +(x− ...
Power series methods of solving ordinary differential equations 501 (iv) The sum of these three terms forms the left-hand side o ...
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