Question

What can be said about the following differential equation? dy 7t It is autonomous, non-separable, linear and non-homogenous It is non-autonomous, non-separable, linear and non-homogenous It is autonomous, separable, linear and homogenous It is autonomous, separable, linear and non-homogenous.

pls choose the answer like a,b,c,d for these 5 multichoice question don't mind what i choose

Answer 1

Linear differential equation dx/dt + P(x) = Q(t) So, By comparing with the standard form dy/dt - 3y = e^7t = > D.E is linear if Q(t) NotEqual 0 then D.E. is non homogenous Here Q9x) = e^7t NotEqual 0 So, D.E. is non homogenous = > D.E. is linear non-homogenous Now, if any D.E. is separable them it can be written in the form dy/dt = f(t) h(y) Here dy/dt = 3y + e^7t which can't be written in the form of f(t) h(x) So, D.E. is non-separable Now, D.E is autonomous if dy/dt = f(y) Now, dy/dt - 3y = e^7t 1 so I.F. e^integral -3 dt = e^-3t multiply by i.F in 1 we get, d/dt (e^3t middot 4) = e^4t = > y middot e^-3t = integral e64t dt = e^4t/4 + c = > y = e^7t/4 + c e^3t So, dy/dt = 7/4 e^7t + 3 c e^3t NotEqual f(x)