Question

in finding the solutions of differential equations by P, Q, method (dy/dx +Py = Q) , when the integrating factor = e ( to the power , integration of P.dx) is it necessary that Q must be in terms of x variable only? Or can terms with y variable be also present?

Answer 1

When solving an equation with integrating factors, the ODE must first be in "standard linear form." That requires that P and Q both be functions of the independent (x) variable only.
standard form:
\[\frac{ dy }{ dx } + p(x)y(x) = q(x)\]

Answer 2

Thank you for your time.
I did not understand 'y(x)" part of your explanation.
Will you please explain again?
What is ODE?
When the differential equation is in linear form with the term dy/dx is it compulsory that P and Q are functions of x ONLY ?
Does this mean that when dx/dy term is present,P and Q must be variables of y?

Answer 3

y(x) means that y is a function of x alone.
ODE is 'ordinary differential equation'.
Yes, P and Q have to be functions of x only, for such an equation to be a Linear Differential equation.
you are right, when it has dx/dy, P and Q must be functions of y alone.

Answer 4

it can be BERNOULLI'S =N of form
\[\frac{ dy }{ dx }+P(x)=Q(x)y ^{n}\]
where LHS is a function of x and y