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Chapter 4: Q3E (page 199)

In Problems 1 through 4, use Theorem 5 to discuss the existence and uniqueness of a solution to the differential equation that satisfies the initial conditions y(1)=Yo,y'(1)=Y1, where Yoand Y1are real constants.

t2y''+y=cost

Short Answer

Expert verified

The differential equation has a unique solution in 0<t<鈭�.

Step by step solution

01

Find the value of p(t), q(t), g(t)

The given differential equation is t2y''+y=cost.

It can be written as y''+y't2=costt2.

So,p(t)=0,q(t)=1t2,g(t)=costt2

02

Check the result

Here p(t), q(t),g(t) is continuous functions in the interval -鈭�<t<0鈥�and鈥�0<t<鈭� and the pointt0=1 in the continuity interval 0<t<鈭�.

Therefore, the differential equation has a unique solution is 0<t<鈭�.

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