A second order Caushy - Euler equation having a solution y(x) = x²cos(4Inx) is %3D x²y" + 5xy + 16y = 0 x²y" + 5xy' 0 O %3D x²y" +4xy' + 20y = 0 x²y" + 20y = 0O %3D x²y" – 5xy' + 20y = 0 O x²y"-5xy' +24y 0 O %3D x²y" + 5xy' + 20y = 0 O %3D

1. Match each function with its equation on the next page. Then identify which function pairs are reciprocals. -2- 2- -4 -2 0 -2 2. -2 b) 2. 2. 4 -2 0 -2 -2- 2)

Show that zxx + Zyy O from the equation z = In(x² + y²) %3D

7. Find 27 a²z xy of z = 2x³ - 5xy²

Q5.4 Let Lcg) (^) 4y) =an you? (x) where the be the following: (n-1) nth + + an-ly Aos Air order Suppose you know 9 2x I 4(y) Ly) = sẻ cosxxẻ xe an linear 2x 2x that: Ly()] = 3 xe when when L[Y₂ (x)] = 7e²x sinxe L[y₂ (x)] = 6e ²x cos x when 2x our e (x) (*) + constants Consider differential equation Find a particular solution be a symbolic function of x any decimals. contain any (x) + a₂ y(x) ал 9, (х) = 6 же ча в 4₂(x) = 14e²x cos x уг 2x 22 93 (x) = 54e²x Cos x + 108€ ²³" sinxe to (*). Your answer should and must not

3. Find the correct form of x(s) and stop. State x(s) as a single fraction. x" +7x' + 6x = 2t x(0) = -1, x'(0) = 2

Determine the solutions of the following first-order ODEs:  Tip:y'=xv'+v          y=xv * xv'+v=(4xv-3x)/(2x-xv) >>>> xv'+v=(2v-3)/(1-v)>>>xv'=(2v-3-v+v²)/(1-v) *  ANSWER CORRECT :|y − x| = C|y + x|3  Any different answer will be considered wrong

x²+1 * 11. Find x'+3x+1

Determine the solutions of the following first-order ODEs:  Tip:y'=xv'+v          y=xv * xv'+v=(4xv-3x)/(2x-xv) >>>> xv'+v=(2v-3)/(1-v)>>>xv'=(2v-3-v+v²)/(1-v) * >>>>>>>>>**  ANSWER CORRECT :|y − x| = C|y + x|3 **<<<<<< *****The answer   |y − x| = C|y + 3x|5   is considered wrong ( will not be accepted as it will be considered wrong)*****

(02-60,9)y=e" (x+2) 3X %3D