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Chapter 3: Q 3.7-1E (page 139)

Determine the recursive formulas for the Taylor method of order 2 for the initial value problemy'=cos(x+y),y(0)=蟺.

Short Answer

Expert verified

yn+1=yn+hcos(xn+yn)-h22sin(xn+yn)(1+cos(xn+yn))

Step by step solution

01

Find the value of f2(x,y)

Here y'=cos(x+y),y(0)=蟺

Apply the chain rule.

f2(x,y)=鈭�f鈭�x(x,y)+鈭�f鈭�y(x,y)f(x,y)

Since f(x,y)=cos(x+y)

鈭�f鈭�x(x,y)=-sin(x+y)鈭�f鈭�y(x,y)=-sin(x+y)

So, the equation is f2(x,y)=-sin(x+y)(1+cos(x+y))

02

Apply the recursive formulas for order 2

The recursive formula is

xn+1=xn+hyn+1=yn+hf(xn+yn)+h22!f2xn+yn+....hpp!fp(xn+yn)

for order p = 2 then

xn+1=xn+hyn+1=yn+hcos(xn+yn)-h22sinxn+yn(1+cos(xn+yn))

Where starting points are xo=0,y0=蟺.

Hence the solution isrole="math" localid="1664314543589" yn+1=yn+hcos(xn+yn)-h22sin(xn+yn)(1+cos(xn+yn))

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