Question

3. The following differential equation is known as the logistic growth equation: y = ry(1-1) where r, k are positive real constants. (a) Note that the logistic growth equation is separable. Use separation of variables to solve the logistic growth equation when r = 1 and k = 2. That is, solve the separable equation: State your solution explicitly. (b) Note that the logistic growth equation is also a Bernoulli equation: y - ry=- C) Solve the logistic growth equation as a Bernoulli equation when r = 2 and k = 2. That is, solve the Bernoulli equation: y – 2y = -4y2 State your solution explicitly. (c) Write 1-2 sentences comparing your work for parts (a)-(b). Did you find one method easier than the other? (d) Use either separation of variables or Bernoulli equations to solve the general version of the logistic equation. y = ry (1-1) State your solution explicitly. Your answer will involve the constants r, k.

Answer 1

I am giving step by step solution of all the four parts. Hope this solution will be helpful. Thank you.

solution :- y'ary (1-44) when 8=1 and K = 2 then, 2 = y - y2 42 2 y' = y(1-2 » y Y RY' Qy_y2 7 & (2Y - 42) -7 Ź (27-42) مه = dy doa =7 dx -17 dy = 24-y2 From this line thro Variables separable. we can see that 2 and as are => S dy ay-y2 - Sa dx . - Sade s dx. dy y (2-4) Y + (2 = 4) y (2-4) I 7 7 dy

=> Julys dy + f f I dy = x + 2ne ntlne => lnc. => In 12-41 + enly I In ly(2-4) => en jy (2-2) y (2-4) => e2 C => y (2-4) ce （ 1 ) > .24 -y2 cen ㅈ y2-2y = - ce² 7 y ² - 2y + 1 l-cel 2 => Y-1)2 1-cé cel yol =7 2 l-ceny 1+ (1-ce?) explicit solution 2 => 1 y This is required (b) y'ury = - ( 2 ) y ² when 8 = 2 and K =1 then, y'-24 = - 4 y2

y'a ay = - 442 Cj༡ A differential equation in the form dy + P(x) y = f(x) yn then this equation is called bernoulli ervation Hence in Coll - 2 P(79) f(n) S -4. And n = 2. in this case we need to substitute V 1-n y't => 1 - 2 V y y/ =7 V st as -12 7 dy da V dv aa Then equation i becomes, dy r2 da 2. I v -4.1 V2 => I dr 4 r dx + 2 V

=> dv +2 V = 4 dx dv da 4- 21. => dv 4-2V =/dx (dx -2% de dr (V-2) dx. z 2 dr (r-2) da. => lon Iv-21 - 2x + lnc. => en IV-a - 23 V-2 7 V-2 - 2x е. V-2 -23 с. е. = => ra -20 2tce -2x 2+ce ㅋ ast =7 as 2+ceax This is explicit solution.

Now we we can (C) if compare the work in then part and (b) ca) say that the method method of Bernoulli part (6) is easy to get a explicit solution. little easy can if you notice you see that in part (a) get the implicit solution in equation as we ice x لے لے we some y (2-4) then further Steps to make it explicit but in the explicit solution directly, d) part (b) we are getting y' = ry ( 1-4 / 4 I am equation using Bernoulli solie the general version, to ( y = ry (1-72 ry - 4² Y y-ry=t& y² - 00 1-2 let v y at

7 as T 1 =7 dy an -1/2 dv ㅗ v2 dore (1) be comes. o -I do da l Exto v2 -7 et ep Xy ку -7 dr da to - ४ К. dr 7 Р (x - * к enx -e => d v da K d ar - -> K (1-kv). (kr-1) - . 8 k dr da dv (KV-) => =Y dr cr- xp *s da K -ret enc. en lr-H In 1 (n= -8x = V- -7X се. 4. V Å tc ra t ce

=> k tcer > ㅋ .de tce k (ktc ** =>y (1+kcette This is required general solution explicit form. in Here in all the cases c is integral constant.