How do I solve the differential equation xy'−y=3xy,y_1=0?

Ratuiszt

Ratuiszt

Answered question

2022-09-30

How do I solve the differential equation x y - y = 3 x y , y 1 = 0 ?

Answer & Explanation

xgirlrogueim

xgirlrogueim

Beginner2022-10-01Added 13 answers

This is a separable equation. We can rewrite the equation as it follows:
x   y - y = 3 x y . Separating the x's and y's, we have that
x y = y + 3 x y
x y = ( 1 + 3 x ) y
y y = 1 x + 3
Now that variables are separated, we can integrate both sides with respect to x:
y y   d x = ( 1 x + 3 )   d x
Considering that y' dx=dy and that the integral of a sum is the sum of the integrals, we have
d y y = 1 x   d x + 3   d x
Solving the ingrals, we have that
log ( y ) = log ( x ) + 3 x + c
Solving for y:
y ( x ) = e log ( x ) + 3 x + c = e log ( x ) e 3 x e c = x e 3 x e c
To determine the value of c, we should use the condition y(1)=0 (if this is what you meant with y 1 = 0 in your question).
I'm a little puzzled with that one, because evaluating y for x=1 we have e 3 e c , and this is never zero for c R . I hope someone can correct me if I said something wrong, or that you can correct the question if the mistake was there.

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Differential Equations

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?