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Chapter 6: Q4E (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.

xx+1y'''-3xy'+y=0y-12=1,鈥�鈥�y'-12=y''-12=0

Short Answer

Expert verified

Hence, the largest interval for the existence of a unique solution on (a, b) to the given initial value problem is-1,鈥�鈥�0.

Step by step solution

01

Step 1:Solve the given equation

The given equation isxx+1y'''-3xy'+y=0.

Divide both sides by x(x+1) in the above equation,

y'''-3x1xx+1y'+1xx+1y=0

Simplify the above equation,

y'''-3x+1y'+1xx+1y=0

Compare with the standard form of a linear differential equation,

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

One has,qx=-3x+1,鈥�鈥�rx=1xx+1

02

Step 2:Check the continuity

qx=-3x+1 is continuous for all x鈮�-1.

rx=1xx+1is continuous in x鈮�0,鈥�-1.

03

Step 3:The largest interval (a, b)

Now q and r continuous for allx鈭�-鈭�,鈥�鈥�-1鈭�-1,鈥�鈥�0鈭�0,鈥�鈥�鈭�

And the initial condition is defined atx0=-12

And-12鈭�-1,鈥�鈥�0

Hence, the largest interval for the existence of a unique solution on (a, b) to the given initial value problem is-1,鈥�鈥�0.

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Most popular questions from this chapter

Reduction of Order. If a nontrivial solution f(x) is known for the homogeneous equation

,y(n)+p1(x)y(n-1)+...+pn(x)y=0

the substitutiony(x)=v(x)f(x)can be used to reduce the order of the equation for second-order equations. By completing the following steps, demonstrate the method for the third-order equation

(35)y'''-2y''-5y'+6y=0

given thatf(x)=ex is a solution.

(a) Sety(x)=v(x)exand compute y鈥�, y鈥�, and y鈥�.

(b) Substitute your expressions from (a) into (35) to obtain a second-order equation in.w=v'

(c) Solve the second-order equation in part (b) for w and integrate to find v. Determine two linearly independent choices for v, say, v1and v2.

(d) By part (c), the functions y1(x)=v1(x)exand y2(x)=v2(x)exare two solutions to (35). Verify that the three solutions ex,鈥�y1(x), and y2(x)are linearly independent on(-鈭�,鈥�鈭�)

In Problems 1-6, use the method of variation of parameters to determine a particular solution to the given equation.y'''+y'=sec胃tan胃,0<胃<蟺/2

In Problems 1-6, use the method of variation of parameters to determine a particular solution to the given equation.

y'''-3y''+4y=e2x

Find a general solution for the differential equation with x as the independent variable.

y'''-3y''-y'+3y=0

Find a general solution to the givenhomogeneous equation.

(D鈭�1)3(D鈭�2)(D2+D+1)(D2+6D+10)3[y]=0
See all solutions

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