How would you solve the next ODE? (dy)/(dt) =(at + by + m)/( ct + dy + n), where a,b,c,d,m,n are constants and ad=bc.

Kayley Dickson 2022-11-21 Answered
Separable Differential Equation
How would you solve the next ODE?
d y d t = a t + b y + m c t + d y + n ,
where a , b , c , d , m , n are constants and a d = b c
Corrected.
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

Taniyah Lin
Answered 2022-11-22 Author has 14 answers
Let u = a t + b y then c t + d y = K u for some constant K (as a d = b c). Now d u d t = a + b d y d t and d y d t = a b + 1 b d u d t . Next we have d y d t = u + m K u + n so a b + 1 b d u d t = u + m K u + n . And this can be solved by separation as it is autonomous.
Did you like this example?
Subscribe for all access

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2022-11-10
Solve separable differential equation d y d x = 2 y 2
Using the initial condition y ( 2 ) = 3, find y ( 1 )
asked 2022-10-27
Representing differential equations explicitly and implicitly
For the past two weeks I have been dealing with differential equations and I have stumbled upon a seemingly simple problem. Suppose I have a separable differential equation d y d x = y ( x ). This simply tells that the function value has to equal to the slope at a particular x value. The solution is therefore simply e x . So far so good. Now suppose I have similiar equation, namely d y d x = x 2 . Now this tells me that the slope of the function at a particular x has to equal x 2 . Hence, the solution is simply the integral, namely 1 3 x 3 . One does not even need the conventional methods to solve separable differential equations.
However, let me now define y ( x ) = x 2 . The two differential equations (and hence the solutions) should be now equivalent. As we have seen, that's not the case. I can sense that I am commiting a mathematical fallacy, however, I do not know what seems to be the problem when I solve the differential equation implicitly rather than explicitly as stated above. Could anyone help me please?
asked 2022-08-17
Solve the following differential equation using separation of variables
f ( x ) + 2 x [ f 2 ( x ) f ( x ) ] = 0
asked 2022-11-19
Separable Differential Equation dy/dt = 6y
The question is as follows:
d y d t = 6 y
y ( 9 ) = 5
I tried rearranging the equation to d y 6 y = d t and integrating both sides to get ( 1 / 6 ) ln | y | + C = t. After that I tried plugging in the 9 for y and 5 for t and solving but I can't quite seem to get it.
asked 2022-10-28
Separable but not exact equation
In class, my professor stated that all separable equations are exact, and we even proved it for homework, but I think I found an equation that is separable but not exact:
( x ln y + x y ) + ( y ln x + x y ) y = 0
My Work:
M = x ln y + x y M y = x y + x N = y ln x + x y N x = y x + y M y N x
But
x ( ln y + y ) + y × y × ( ln x + x ) = 0 x ( ln y + y ) = y × y × ( ln x + x ) x ln x + x = y × y ln y + y
Separated
So whats up? Am I doing something wrong?
asked 2022-11-23
Correct approach to this homogeneous differential equation
I am trying to find the general solution to the following equation, but the integral at the end is very complicated and leads me to believe I may have made a mistake somewhere.
x y 2 d y d x = y 3 + x y 2 x 2 y x 3
Which, by the substitution z = y x , can be rearranged into the equation
x d z d x = 1 1 z 1 z 2
This is a separable equation, which I separated into
1 1 1 z 1 z 2 d z = 1 x d x
The right-hand side is easy to solve, but the left-hand integral is giving me trouble. Assuming I did the steps leading up to it correctly, the integral has me stumped. Even WolframaAlpha is unhelpful. My first thought would be to try a partial fraction, but after a few attempts it does not seem to work.
Am I approaching this differential equation correctly? Is there an error I haven't caught?
asked 2022-08-16
Solving a separable differential equations
We have an equation:
d P d t = k P ( 1 P M ) ,
and we need to find P ( t ) given the initial P ( 0 ) given the initial P ( 0 ). Here k is a constant and P represents population, M represents maximum population.
I tried to use separable differential equations, but I am slightly confused. There is a hint:
M P ( M P ) = 1 P + 1 M P ,
and I need to simplify as far as possible. I have tried to use separable equations but I cannot seem to get the hint equation. I think the main issue is trying to get all P's on to one side.
Thank you in advance!