Question

8.1.3 Determine the first three nonzero terms in the Taylor polynomial approximation for the given initial value problem. y' = 8 sin y +2e5Xy(0) = 0 The Taylor approximation to three nonzero terms is y(x) = +..

Answer 1

Answer:

$P(x)=2x+13x^{2}+43x^{3}$

Step-by-step explanation:

We follow the procedure to find each term of the Taylor series:

$y'=8sin(y)+2e^{5x}\\y'(0)=8sin(0)+2e^{5(0)}\\y'(0)=2\\y''=y'8cos(y)+10e^{5x}\\y''(0)=2*8cos(0)+10e^{5(0)}\\y''(0)=26\\y'''=y''8cos(y)-(y')^{2}8sin(y)+50e^{5x}\\y'''(0)=26*8cos(0)-(2)^{2}8sin(0)+50e^{5(0)}\\y'''(0)=258$

Now that we have the first three non-zero terms, we proceed to set the Taylor series:

$P(x)=0+\frac{2}{1!} x+\frac{26}{2!} x^{2}+\frac{258}{3!} x^{3}$

Then we proceed to solve the polynomial:

$P(x)=2x+13x^{2}+43x^{3}$