At s=1/4, the pieces Ax and B(1-x) meet continuously. With B nonzero, what is A/B?

For -u'' on [0,1] with zero endpoint values, what is G(1/4,1/2)?

• 1/8
• 3/8
• 1/4
• 1/2

For -u''=f on [0,1] with u(0)=u(1)=0, which formula is the Green's function?

• G=s(1-x) for x<=s; G=x(1-s) for x>=s
• G=x(1-s) for x<=s; G=s(1-x) for x>=s
• G=x(1-s) for x<=s; G=-s(1-x) for x>=s

For the taught separated boundary problem, how are the two homogeneous pieces used to construct G?

• Meet each endpoint condition, match values, then set the slope jump.
• Meet each endpoint condition, match values, then make both slopes equal.
• Meet only the left endpoint condition, then keep one line across the interval.

For the two-zero-endpoint kernel G(x,s)=s(1-x) on x>s, fix s=1/5. What is its right-side x derivative? Give a decimal.

For -2u''=f on [0,1] with zero endpoint values, put a unit source at s=1/4. The left x slope of G is 3/8. What is the right slope? Give a decimal.

For -u''=f on [0,1] with u(0)=0 and u'(1)=0, which Green's function meets the conditions and unit-source jump?

• G=s for x<=s; G=x for x>=s
• G=-x for x<=s; G=-s for x>=s
• G=x for x<=s; G=s for x>=s
• G=x for x<=s; G=s(1-x) for x>=s

The taught Green's function for -u'' on [0,1] with u(0)=u(1)=0 satisfies G(x,s)=G(s,x).

For -u'', replacing u(1)=0 by u'(1)=0 while retaining u(0)=0 changes the Green's function.

For -u''=1 on [0,1] with u'(0)=u'(1)=0, why is there no solution?

• A positive forcing requires positive derivatives at every point of the interval.
• A second-order equation cannot use derivative conditions at its endpoints.
• Constant homogeneous solutions make every forcing incompatible with these conditions.
• Integration gives -u'(1)+u'(0)=1, but the conditions make that left side zero.