Question

Problem 5. (1 point) A Bernoulli differential equation is one of the form +P()y= Q()y" (*) Observe that, if n = 0 or 1, the Bernoulli equation is linear. For other values of n, the substitution u =y- transforms the Bemoulli equation into the linear equation + (1 - x)P(3)u = (1 - x)^(x). Consider the initial value problem ry' +y = -3.xy?, y(1) = 2. (a) This differential equation can be written in the form (*) with P(1) = Q(t) = , and (b) The substitution u = will transform it into the linear equation du de : (c) Using the substitution in part (b), we rewrite the initial condition in terms of rand u(1) = (d) Now solve the linear equation in part (b). and find the solution that satisfies the initial condition in part (c). u(z) = (e) Finally, solve for y y(1) =

Answer 1

xy+y=-3xy , 4(1)=2 sol xy + y = -3xy? xdy + y = -33y? (9 (4)= -333°C ed (1.99 = -3y? Here pcmc) @ This eq. can be written in the form P(x) = 1 / Q(20) = (-3 , and m= 2 with hoe Y = -3y? g-2 [ dy + 1 / 2 y] =-3 yo pe t -3 The substitution a= [y-1 u=yol delle + 1 u=-3 doc-14=3 + O BLI du + Cuza u=y-1 (substitution) uý uc ID= y el

du - 1 4 4:3 , which is a linear diffeq. PCDC) -, Q(x)=3 Spcoesde 5 - 72 dx e-lux IF :. gen.sd of linea diff.eq. is ue spessore = sa. esperado date untjie = 53. 1 dacte Con = 3 (note U = (Elməc+c) -> * given uc) = 2 :-) 1 : 1631nc18+c) à = 1 (o+C) => c= į u= x(31nx+ 1/2) u= 3x nx+of = 6xlnxto :46%) = [6&ln (20) + x 2

@sol from we have uz (3 lmac+c) but u=hl y's (31n*c+c) s = x(31)x+c) y = xcalmxtc) given Y61)=2 2= Calmisto 2= ; ¢3603+ 2cas :y (3 nx+z) * Balmsetzen 6xclnxta 2 5.9()3 Bollmcastz - 2 6xlnxtx ..(x) = 2 6xln(x)+x