Question

: Solve the following differential equation eigenvalue problems.

a y'' + λy = 0; y(0) = 0; y(4) = 0

b y''+ λy = 0; y(0) = 0; y' (1) − 2y(1) = 0

In problem [a] you may assume that there are no eigenvalues for λ ≤ 0. In problem [b] you will not be able to find the exact eigenvalues. You should find a condition on the eigenvalues of the form f(µ) = 0 where µ 2 = λ. In this case there is an eigenvalue λ < 0. Find as equation for this eigenvalue in the form g(µ) = 0 with −µ 2 = λ and estimate λ by graphing g(µ) and estimating where it is zero.

Answer 1

10) + AY =0 40o,0 , 4 (4)=0 - Y = C, cos dat & co sin dat 10) = 0 = 0 y (9)=0 Ca din 40 =0 sin a no 4 5a = nt - ana nga In dies but y" + 2y=0, 9(0)=0, 9' (1-290)=0 7=0 » y = ax + b y 10) = ² b=0 y (1) - 2901)=0 + 9 -9950 950 » Y=0 u zo say 92-02 en y= de&Ga e at y logo & C + C =0 y'(1) - 8901)=0 -> 06, la Gueº) - (Ate")=0 - 0,26220 ao say out Y = C, cosa + c, sin ux y10130 >0,59, 9'11)-990) zo >>4.0084 - 2 sinu so (M = 2 Tanul In = Eins