Question

Hello, please help with the following problem - I will give positive rating for fast response!

(a) Write down the general form of a SLBV problem including the boundary conditions at the ends (b) Consider the second order boundary value problem -[(1+x?)y'l' = 4y + 1,y(1) = 0, y(0) =0,0 < x 51. Is this a SLBV problem? Explain. (c) Derive the eigenvalues and eigenfunctions of y" + dy = 0, y'(O)= () = y(1)+y'(1) .Is A = 0 an eigenvalue? Justify your answer. (d) Using the eigenvalues and eigenfunctions derived in (c), solve the nonhomogeneous SLDE y" + 3y + (1 - x) = 0,0 < x <1 with the same BC as in (c).

Answer 1

10 The general form of a SL differential equation on a fiorite interval Paib] with homogene mixed boundary condition is - [Pa dy + q (a) y = 7W(2) y = x 6 [a, b] with Boundary conditions Ca 4 (a) + day (a) =0 Gy (6) + 2 y (b) = 0 P (2, W (270 for XE [a, b] 8 - [C+ x)² y] =ay +1 &Y (1) = 0, 4 (0)=0 olx 21. or – of [+2² de Jy + 0 = 2.1.4+1 Compaining with we get P(G) = (1+02 ; q (x) = 0; W(x) = 1 Since there exist extra 1, it is not a SL Boundary value problem. @ Given S-L problem is y tay = 0 ;4(o)=0=4 (1)+y (1) m7a=0 solution is The auxitary equation is Case-I o when a=0, the y = ex (a +Gx) = G+Cox & Since m= 0,0 y (a) = oto

Giren Boundary condition is 4(o)=0 and 4 (1) 5+ y(1)=0 put this in ③ we get o=cs and C te to=0 or, 2c +c=0 or 9=0 in the solution is y=0 ; which is a troivial solution, Cass-I for aco, the solution as y=cet-2x + C e Tax where (-2) and 5-7 are positive. How apply Y'(x) = oA) e x a Tax putting boundary conditions 90 © 0 = 50 C 1-2 or, 6C) = 0 or G=ca and get + e + gone #-cot-aéta* = 0 or, Get [1 +5-2] * = C eta era-1) puiting a = ca in above equation We get the troivial solution G=6=0

Case-llo when ayo, the solution is y = ao cos six + C sin saa and y'= dos-c, Sindax + C Costa x} When yo)=0, we have odno &, o=0 to And The Solution & seduce y = C, cos dix ; Whereby - © f (x) = - Ine Sintax But given y' (1) & 4 (1) = 0 put this in @ ② and - The Sanda + G costa = 0 or a [ costa da sam nJ=0 To obtain nontrivial Solution, we must have to then costa - San do =0 be.cosa sma or / 3 = tan. In are Therefore, the eigen values are an = tem for n= 1,2,3,.. and agen-functions will be L Y = An cos sana on=1, 2, 3,... and o is not a ligen value.