Higher Engineering Mathematics

502 DIFFERENTIAL EQUATIONS i.e.y=a 0 x 1 2 { 1 +x+ x^2 (2×3) + x^3 (2×3)×(3×5) + x^4 (2× 3 ×4)×(3× 5 ×7) + ··· } (27) Sincea 0 i ...

POWER SERIES METHODS OF SOLVING ORDINARY DIFFERENTIAL EQUATIONS 503 I whenr=2,a 4 = 2 a 2 3 × 4 = 4 a 0 4! Hence,y=x^0 { a 0 +a ...

504 DIFFERENTIAL EQUATIONS Use the Frobenius method to determine the general power series solution of the differen- tial equat ...

POWER SERIES METHODS OF SOLVING ORDINARY DIFFERENTIAL EQUATIONS 505 I For the term inxc+r, ar(c+r−1)(c+r)+ar(c+r)+ar− 2 −arv^2 = ...

506 DIFFERENTIAL EQUATIONS The complete solution of Bessel’s equation: x^2 d^2 y dx^2 +x dy dx + ( x^2 −v^2 ) y=0 is: y=u+w= Axv ...

POWER SERIES METHODS OF SOLVING ORDINARY DIFFERENTIAL EQUATIONS 507 I = −a 2 23 (v+2) = − 1 23 (v+2) − 1 2 v+^2 (v+2) = 1 2 v+^ ...

508 DIFFERENTIAL EQUATIONS Another Bessel function It may be shown that another series forJn(x)is given by: Jn(x)= (x 2 )n{ 1 n! ...

POWER SERIES METHODS OF SOLVING ORDINARY DIFFERENTIAL EQUATIONS 509 I Evaluate the Bessel functionsJ 0 (x) andJ 1 (x) whenx=1, ...

510 DIFFERENTIAL EQUATIONS = −a 2 (k+3)(k−2) (3)(4) = −(k+3)(k−2) (3)(4) . a 0 [−k(k+1)] (1)(2) = a 0 k(k+1)(k+3)(k−2) 4! Forr=3 ...

POWER SERIES METHODS OF SOLVING ORDINARY DIFFERENTIAL EQUATIONS 511 I Rodrigue’s formula An alternative method of determining Le ...

Differential equations 53 An introduction to partial differential equations 53.1 Introduction A partial differential equation is ...

AN INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS 513 I 53.3 Solution of partial differential equations by direct partial integr ...

514 DIFFERENTIAL EQUATIONS The partial differential equation ∂^2 φ ∂x^2 + ∂^2 φ ∂y^2 + ∂^2 φ ∂z^2 =0 is calledLaplace’s equation ...

AN INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS 515 I Solve ∂^2 u ∂x∂t =sin(x+t) given that ∂u ∂x = 1 whent=0, and whenu= 2 ...

516 DIFFERENTIAL EQUATIONS Worked Problem 4 will be a reminder of solving ordinary differential equations of this type. Problem ...

AN INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS 517 I ButX=0atx=0, hence 0=A+Bi.e.B=−Aand X=0atx=L, hence 0 =AepL+Be−pL=A(epL− ...

518 DIFFERENTIAL EQUATIONS andBn (cnπ L ) is twice the mean value of g(x)sin nπx L betweenx=0 andx=L i.e. Bn= L cnπ ( 2 L )∫L 0 ...

AN INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS 519 I Separating the variables gives: X′′ X = T′′ T Let constant, μ= X′′ X = ...

520 DIFFERENTIAL EQUATIONS Now try the following exercise. Exercise 202 Further problems on the wave equation An elastic string ...

AN INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS 521 I If lnT=−p^2 c^2 t+c 1 then T=e−p (^2) c (^2) t+c 1 =e−p (^2) c (^2) t ec ...