Higher Engineering Mathematics

482 DIFFERENTIAL EQUATIONS Table 51.1 Form of particular integral for different functions Type Straightforward cases ‘Snag’ case ...

SECOND ORDER DIFFERENTIAL EQUATIONS (NON-HOMOGENEOUS) 483 I (ii) Substituting m for D gives the auxil- iary equationm^2 − 3 m=0. ...

484 DIFFERENTIAL EQUATIONS 9 d^2 y dx^2 − 12 dy dx + 4 y= 3 x−1; whenx=0, y=0 and dy dx =− 4 3 [ y=− ( 2 +^34 x ) e 2 3 x+ 2 + ...

SECOND ORDER DIFFERENTIAL EQUATIONS (NON-HOMOGENEOUS) 485 I the P.I., v=kxe 3 2 x (see Table 51.1(c), snag case (i)). (iv) Subst ...

486 DIFFERENTIAL EQUATIONS 3. d^2 y dx^2 + 9 y=26e^2 x [y=Acos 3x+Bsin 3x+2e^2 x] 9 d^2 y dt^2 − 6 dy dt +y=12e t 3 [ y=(At+B) ...

SECOND ORDER DIFFERENTIAL EQUATIONS (NON-HOMOGENEOUS) 487 I Using the procedure of Section 51.2: (i) d^2 y dx^2 + 16 y=10 cos 4x ...

488 DIFFERENTIAL EQUATIONS A differential equation representing the motion of a body is d^2 y dt^2 +n^2 y=ksinpt, wherek,nandp ...

SECOND ORDER DIFFERENTIAL EQUATIONS (NON-HOMOGENEOUS) 489 I (vi) The general solution,y=u+v, i.e. y=Ae^2 x+Be−^3 x− 2 x −^13 +7 ...

490 DIFFERENTIAL EQUATIONS Now try the following exercise. Exercise 193 Further problems on second order differential equations ...

I Differential equations 52 Power series methods of solving ordinary differential equations 52.1 Introduction Second order ordin ...

492 DIFFERENTIAL EQUATIONS For example, if y=4 cos 2x, then d^6 y dx^6 =y(6)=4(2^6 ) cos ( 2 x+ 6 π 2 ) =4(2^6 ) cos(2x+ 3 π) =4 ...

POWER SERIES METHODS OF SOLVING ORDINARY DIFFERENTIAL EQUATIONS 493 I (a)y(8)wheny=cos 2x (b)y(9)wheny=3 cos 2 3 t [ (a) 256 c ...

494 DIFFERENTIAL EQUATIONS Differentiating each term ofx^2 y′′+ 2 xy′+y= 0 ntimes, using Leibniz’s theorem of equation (13), giv ...

POWER SERIES METHODS OF SOLVING ORDINARY DIFFERENTIAL EQUATIONS 495 I Determine the 4th derivative of:y= 2 x^3 e−x [y(4)=2e−x( ...

496 DIFFERENTIAL EQUATIONS (iv) Maclaurin’s theorem from page 67 may be written as: y=(y) 0 +x(y′) 0 + x^2 2! (y′′) 0 + x^3 3! ( ...

POWER SERIES METHODS OF SOLVING ORDINARY DIFFERENTIAL EQUATIONS 497 I From the given differential equation, y′′+y′+xy=0, and, at ...

498 DIFFERENTIAL EQUATIONS 52.5 Power series solution by the Frobenius method A differential equation of the form y′′+Py′+ Qy=0, ...

POWER SERIES METHODS OF SOLVING ORDINARY DIFFERENTIAL EQUATIONS 499 I the value ofcis obtained. From equation (18), sincea 0 =0 ...

500 DIFFERENTIAL EQUATIONS Leta 0 =Ain equation (21), anda 0 =Bin equation (22). Also, if the first solution is denoted byu(x) a ...

POWER SERIES METHODS OF SOLVING ORDINARY DIFFERENTIAL EQUATIONS 501 I Thus, whenr=1, a 1 = a 0 1(2+1) = a 0 1 × 3 whenr=2, a 2 = ...