Higher Engineering Mathematics, Sixth Edition
502 Higher Engineering Mathematics Thus, whenr=1,a 2 = a 1 ( 2 × 8 ) = a 0 ( 2 × 5 × 8 ) sincea 1 = a 0 5 whenr=2,a 3 = a 2 ( 3 ...
Power series methods of solving ordinary differential equations 503 −a 0 xc+^1 −a 1 xc+^2 −a 2 xc+^3 −a 3 xc+^4 −···−arxc+r+^1 − ...
504 Higher Engineering Mathematics = ar− 1 2 r^2 − 1 2 − 1 2 −r+ 1 = ar− 1 2 r^2 −r = ar− 1 r(2r−1) Thus, whenr=1,a 1 = a 0 1 ( ...
Power series methods of solving ordinary differential equations 505 Substitutingyandy′′into each term of the given equationy′′− ...
506 Higher Engineering Mathematics thecasewhenthetwovaluesofcdifferbyaninteger(i.e. whole number). From theabove three workedpro ...
Power series methods of solving ordinary differential equations 507 i.e. y=a 0 xc+a 1 xc+^1 +a 2 xc+^2 +a 3 xc+^3 + ···+arxc+r+· ...
508 Higher Engineering Mathematics which is valid providedvis not a negative integer and whereAis an arbitrary constant. Whenc=− ...
Power series methods of solving ordinary differential equations 509 + (x 2 )−v{ 1 ( 1 −v) − x^2 22 (1!)( 2 −v) + x^4 24 (2!)( ...
510 Higher Engineering Mathematics i.e. y=AJv(x)+BJ−v(x) =A (x 2 )v{ 1 (v+1) − x^2 22 (1!)(v+2) + x^4 24 (2!)(v+4) −··· } +B ...
Power series methods of solving ordinary differential equations 511 Now try the following exercise Exercise 197 Further problems ...
512 Higher Engineering Mathematics from which, ar+ 2 = ar [ (c+r− 1 )(c+r)+ 2 (c+r)−k^2 −k ] (c+r+ 1 )(c+r+ 2 ) = ar[(c+r)(c+r+ ...
Power series methods of solving ordinary differential equations 513 Hence, P 2 (x)=− 1 2 ( 1 − 3 x^2 ) = 1 2 (3x^2 −1) Problem 1 ...
514 Higher Engineering Mathematics term inx^5. ⎡ ⎢ ⎢ ⎢ ⎢⎢ ⎣ (a)y=a 0 +a 1 ( x+ x^3 3 + x^5 5 +··· ) (b)y=a 0 { 1 − 3 x^2 } +a 1 ...
Chapter 53 An introduction to partial differential equations 53.1 Introduction A partial differential equation is an equation th ...
516 Higher Engineering Mathematics and integrating ∂u ∂x partially with respect toxgives: u= ∫ [3x^2 sin2y+f(x)]dx =x^3 sin2y+(x ...
An introduction to partial differential equations 517 Problem 3. Verify that φ(x,y,z)= 1 √ x^2 +y^2 +z^2 satisfies the partial d ...
518 Higher Engineering Mathematics Solve ∂^2 u ∂x∂t =sin(x+t)given that ∂u ∂x = 1 whent=0, and whenu= 2 twhenx= 0. [u=−sin(x+t ...
An introduction to partial differential equations 519 Worked Problem 4 will be a reminder of solving ordinary differential equat ...
520 Higher Engineering Mathematics have solutions:X=Aepx+Be−pxand T=Cecpt+De−cptwhereA,B,CandDare constants. ButX=0atx= 0 ,hence ...
An introduction to partial differential equations 521 andBn (cnπ L ) is twice the mean value of g(x)sin nπx L betweenx=0andx=L i ...
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