Question

2. Solve (i) Find the general solution of y" + y' = 0. Here, don't have to check the Wronskain not zero for the fundamental set of solutions. (ii) For the initial conditions y(e) = e, y'(e) = sin(e), y" (e) = e and y"(e) = e sin(e), show that there is a unique solution satisfying the conditions. (Hint: you don't have to explicitly solve for it)

Answer 1

we have Page 1 @ y" #y" = 0 - 0 or 3 you becomes - $340) = 0 ( 3 + y =o 2²€ AI) = 0 D2=0 or P + ZO D=0,0,– 1 then solution of ② as y = (a + x) et ce a go = c + GR & Ge thence genental Solution of ② is given as then You = G + 6x + ezek /

I we have y Initial condition = 0 - i Je = sine ye - e - ri -01) Now from Y'y = @ enged - @ yu = C,+Garces 1 Now Applying the Gig (i) and H) x'h @ y = zed Initial condition 0,+ © by i ga e ze f ined constant Similary we can find e, a ne can also get , & Co as fixed constat that is unique. Hence there is a unique solution, which satisfies above snitial Condition. Hence proced.
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