First order ODE: \(\displaystyle{a}{\left({y}\right)}{y}^{'}+{y}={b}{\left({t}\right)}\) Please, give advise or reference

Terrence Riddle

Terrence Riddle

Answered question

2022-04-05

First order ODE:
a(y)y+y=b(t)
Please, give advise or reference for solving first order ODE:
a(y)y+y=b(t), where a, b are known function. It would be better to find just one solution.

Answer & Explanation

Alan Bean

Alan Bean

Beginner2022-04-06Added 10 answers

Step 1
There isn't a general form for the solution but one can study some special cases :
1. Cas b is constant the the equation become with separated variables: If:
a(y)y+y=b(t)=λ
then: a(y)λydy=dx=x+C
2. Cas a and b of the same form a(t)=mt+n and b(t)=pt+q:
In this case we have :
y'=pty+qmy+n
Step 2
With variable change of type:
y1=y+u  and  t1=t+v
it becomes homogeneous of type:
dy1dt1=dαy1+βt1γy1+λt1

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