Question

5.

Given that \(y=x\) is a solution of \(\left(x^{2}-x+1\right) y \prime \prime-\left(x^{2}+x\right) y \prime+(x+1) y=0\), a linearly independent solution obtained by reducing the order is given by

\(y=e^{x}(x+1)\)

\(y=e^{x}(x-1)\)

None of them

\(y=x^{2} e^{x}\)

\(y=x e^{x}\)

6.

 If the functions y = x and y = xe^x are linearly independent solutions of the non-homogeneous second-order linear differential equation with variable coefficients second-order linear differential equation with variable coefficients

\(x^{2} y \prime \prime-x(x+2) y \prime+(x+2) y=x^{3}\), its general solution is given by

None of them

\(y=C_{1}+C_{2} x e^{x}+x^{2}\)

\(y=C_{1} x^{2}+C_{2} x e^{x}-x^{3}\)

\(y=C_{1} x+C_{2} x e^{x}-x^{2}\)

\(y=C_{1} x+C_{2} x^{2} e^{x}-x^{3}\)

Answer 1

option b is true for answer 5 and option a is true for answer 6

Answer 2

(02-271) y' - (2+) y +on+ 1) y = 0 - U one solution is y=n. y = ur. .y'= untu y = uinta' ta' un +2m' from (n. (? n +1) Cue">+2M')-(2+) (au'r tu) + (a1 +1) urco nu" +222 "-²""-22 i + 2 utru' nu'nu-murnu trhu truco = (a}-+ ) u" +(-+2-2n+2) el antha-22+2 no u" + Let, u=w :m"=wi -2 +2² -22 +2 wilt +2 dw -2 +22-22+2 dar 2-2²th dao 2-2² +22-2 da a U -t2 L nr 2 + 2)dor 21-2²+21 alnews Ja ln (w)= /dn dan + n-2 2(22_2+1)

la(w) = nt da -20%-n+1) +7(222-1) 2(2-2+1) 22-1 > ln (w)= nt + ant1 ht) da . > ln(ol =n-2 ln(2) + ln(2²-2+1) + ln (9) In(a) en ( 9 (222-2+1) er **+1) €*) &w= a(h+ben 22 2. u'ae (nontll e » du = ja Chant) e da ofdm = aferde-celle e da Gent aen + ₂ y=arena en tan =c(2-1) en tar other solution is y= (2-1) en Answer is y=(2-1) en

6) 22 y" - 2 (2+2) Y' +(2+2) Y =23 yan and y=ner are linearly independent solution. : Yes arteaner .. =nen WCY, 4) = y xlu 1 methoden nen nente" anten theranen ty 9(2)= nen (2) da YA () W(YY) don --|- 4, 8(2) W(Guy) da = I nema dona/endrane .. p = y a ty andaren 2²-2 (ar)n + Care an Amwer is y=artlane nen_2²