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Chapter 6: Q2E (page 326)

Determine the largest interval (a, b) for which Theorem 1 guarantees the existence of a unique solution on (a, b) to the given initial value problem.

y'''-xy=sinxy(蟺)=0,鈥�鈥�y'(蟺)=11,鈥�鈥�y''(蟺)=3

Short Answer

Expert verified

Hence, the largest interval0,鈥�鈥�鈭�.

Step by step solution

01

Solve the given equation

The given equation isy'''-xy=sinx.

Compare with the standard form of a linear differential equation,

y'''+pxy''+qxy'+rxy=sx

Therefore,

rx=-x,鈥�鈥�sx=sinx

02

Step 2:Check the continuity

rx=-xis continuous for all x猢�0.

sx=sinxis continuous in R.

03

Step 3:The largest interval (a, b)

Now overall r and s is continuous in鈭赌鈥�x鈭�0,鈥�鈥�鈭�.

The initial condition is defined atx0=蟺.

And蟺鈭�0,鈥�鈥�鈭�.

Hence, the largest interval0,鈥�鈥�鈭�.

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