Solution Exam 1 (MATH246, Spring 2024)

Problem 1(a)

separable, constant solution

syms y(t) % defines symbolic variable t and symbolic function y(t)

Dy = diff(y);

ode = Dy+t*(y-1)^2==0;

dsolve(ode)

ans = 

We get two solutions: one from separation (for ), and the constant solution .

dsolve(ode,y(2)==-1)

ans = 

solution does not exist for , solution exists for

solution exists for all

Problem 1(b)

ODE is exact:

Find with and : we get .

ode = -4*y+16*t+(2*y-4*t)*Dy==0;

dsolve(ode,'Implicit',true) % solution in implicit form

ans = 

dsolve(ode)

ans = 

dsolve(ode,y(0)==-1)

ans = 

solution exists for , i.e.,

Problem 2(a)

Problem 2(b)

(i) for we get as , as , solution exists for all

(ii) for we get as (solution does not exist for since improper integral is finite)

as , solution exists for

Problem 3(a)

denotes total amount of water in tank at time t,

volume of water in tank at time t is

average concentration in tank is

ODE , initial condition

Problem 3(b)

linear ODE , , , integrating factor

ode = Dy==6-2*y/(10+t);

dsolve(ode)

ans = 

dsolve(ode,y(0)==50)

ans = 

Problem 4