Kindly solve the problem given in the image attached here. Thank you! All the qu

Question

Kindly solve the problem given in the image attached here. Thank you!

All the questions on this homework have to do with the the differential equation y" + 5xy' + y = 0. Is it true that if u(x) is a solution to the differential equation, then u(x) + c is also a solution for all (or some) constants c? Answer the question as completely as you can and justify it as thoroughly as you can 1 Suppose I give you two solutions u1 and u2 Show that for any constants c1, c2, the linear combination u(x) = c1 - u1(x) + c2 middot u2(x) is also a solution to the differential equation.

Explanation / Answer

a) Replacing v(x)=u(x)+c in the above equation you end up with c=0 , indeed :

v''(x)+5xv'(x)+v(x) = (u''(x)+5xu'(x) + u(x)) + c = 0 +c

b)

Suppose u1(x) and u2(x) are solution then for c1,c2 you have :

v(x)=c1u1(x)+c2u2(x)

v''(x)+5xv'(x)+v(x) = c1(u1''(x)+5xu1'(x)+u1(x))+c2(u2"(x)+5xu2'(x)+u2(x)) = 0+0 = 0

So v is also solution

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