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Chapter 1: Q26 E (page 14)

In Problems 23-28, determine whether Theorem 1 implies that the given initial value problem has a unique solution.

dxdt+cosx=sint,x(蟺)=0

Short Answer

Expert verified

The hypotheses of Theorem 1 are satisfied.

The theorem shows that the given initial value problem has a unique solution.

Step by step solution

01

Finding the partial derivative of the given relation concerning y.

Here,ft,x=sint-cosx

and

鈭�f鈭�x=--sinx鈭�f鈭�x=sinx

02

Determining whether Theorem 1 implies the existence of a unique solution or not

Now from Step 1, we find that both of the functionsft,xand鈭�f鈭�x are continuous in any rectangle containing the point 蟺,0, so the hypotheses of Theorem 1 are satisfied. It then follows from the theorem that the given initial value problem has a unique solution in an intervalt=蟺 about of the form 蟺-未,蟺+未, where未is some positive number.

Hence, Theorem 1 implies that the given initial value problem has a unique solution.

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