Using Taylor Series for Initial Value Problems

[roldy]

Homework Statement

I posted this already but decided to revive this thread since I re-worked the problem.

Consider dy/dx=x+y, a function of both x and y subject to initial condition, y(x₀)=y₀.
Use Taylor series to determine y(x₀+\Deltax) to 4th order accuracy.

Initial condition: x₀=0, y(x₀)=1
step size: \Deltax=0.1

Show 5 significant digits in the answer.
Do the calculations for only one step.

Homework Equations

\epsilon=O(\Deltax⁵)

The Attempt at a Solution

dy/dx=f(x,y)

Taylor series:
y(x₀+\Deltax)=y(x₀)+\Deltaxf'(x₀,y(x₀))+
\Delta[/itex]x²1/2!f''(x₀,y(x₀))+\Deltax³1/3!f'''(x₀,y(x₀))+\Deltax⁴1/4!f''''(x₀,y(x₀))+\epsilon+<br /> <br /> Is this correct?<br /> <br /> My solution:<br /> <br /> The derivatives:<br /> f&#039;(x,y)=dy/dx=y+x=1+0=1<br /> f&#039;&#039;(x,y)=d²y/dx²=dy/dx+1=1+1=2<br /> f&#039;&#039;&#039;(x,y)=d³y/dx³=dy²/dx²=2<br /> f&#039;&#039;&#039;&#039;(x,y)=d⁴y/dx⁴=d³y/dx³=2<br /> <br /> y(0+.1)=1+(.1)(1)+1/2!(.1)²(2)+1/3!(.1)³(2)+1/4!(.1)⁴(2)+(.1)⁵<br /> <br /> y(.1)=1+.1+.01+.001/3+.0001/12+.00001=1.11035<br /> <br /> Did I solve this correctly? I want to be able to have something decent when I meet with the professor tomorrow.

 

[PAllen]

Everything looks right. I am not willing to double check all your arithmetic.

 

[roldy]

That's fine. I just wanted to know if I was on the right track. Thanks for taking a look.