What is the Implicit answer for this differential equation? (dy)/(dx)=y^2-4

Milton Anderson 2022-09-14 Answered
I need help with Differential equations.
What is the Implicit answer for this differential equation?
d y d x = y 2 4
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Answers (2)

Leon Webster
Answered 2022-09-15 Author has 17 answers
Step 1
Here is how you solve equations like yours. You are trying to solve d y d x = f ( y ). Rewrite it as d y f ( y ) = d x, integrate on both sides 1 f ( y ) d y = d x = x + c and invert to get expression for y(x). If you have conditions on the value of y(x) as some particular x, like x = 0, then you can determine c as well.
Step 2
Applying this to your example. You have 1 y 2 4 d y = x + c. The integral is 1 4 log ( 2 y 2 + y ) so that 2 y 2 + y = exp ( 4 x + 4 c ) . From here solving for y is straightforward.
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Modelfino0g
Answered 2022-09-16 Author has 1 answers
Step 1
d y d x = y 2 4 d y y 2 4 = d x
Then integrate each side. You can easily use partial fractions on the left hand side.:
d y y 2 4 = d y ( y 2 ) ( y + 2 ) = A y 2 + B y + 2
A y + 2 A + B y 2 B = 1 ( A + B ) y + 2 ( A B ) = 1 ( A + B = 0 , 2 ( A B ) = 1 ) ( A = B 4 B = 1 ) .
So A = 1 4 , B = 1 4
That gives us
1 2 1 y 2 1 y + 2 d y = d x + c
(1) 1 4 ln ( y 2 y + 2 ) = x + C
Step 2
Given your last comment, we can continue as follows from (1):
(2) ln ( y 2 y + 2 ) = 4 x + C
Raising each side as a power of e gives us:
ln ( y 2 y + 2 ) = 4 x + C e ln ( y 2 y + 2 ) = e 4 x + C
y 2 y + 2 = e 4 x e C C
So (3) y 2 y + 2 = C e 4 x
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