Transition in solving differential equations using Method of

gabolzm6d

gabolzm6d

Answered question

2022-04-08

Transition in solving differential equations using Method of Undetermined Coefficient
x2y3xy+3y=lnx

Answer & Explanation

Raina Blackburn

Raina Blackburn

Beginner2022-04-09Added 10 answers

Step 1
The equations defined by operators of the form
L(y)=k=0nakxky(k)=b(x), with akR can be solved applying the substitution y(x)=y(et). Observe that L(y)=b(x) is not in normal form so you have to study the equation in
J=(,0)(0,+)
Consider
x(0,+),\ if y(x)=y(et)=Y(t)
you obtain
Y'(t)=y'(et)et=y'(x)x and
Y (t)=y (et)etet+y(et)et=y (et)e2t+Y(t).
From these relations if follows that
Y'(t)=y'(et)et=y'(x)x
The initial equation is now
Y (t)Y(t)3Y(t)+3Y(t)=log(et), so
Y (t)4Y(t)+3Y(t)=t
which is a second order differential equation with constant coefficients and the general solution is given by Y(t)=Yhom(t)+Yp(t), where Yhom is the solution for the homogeneous problem and Yp(t) is the particular solution.
The characteristic polynomial associated to the homogeneous problem is P(μ)=μ24μ+3μ, whose roots are μ1,2=4±22=2±1, then
Yhom(t)=c1e3t+c2et,  c1,c2
The general particular solution is Yp(t)=at+b, with a,bR to determine.

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