So I need to solve this differential equation
dy/dx = 1/(x+2)
so i solved it like this
dy = 1/(x+2)dx
then i took the integral of both sides
y= ⌠1/(x+2)dx
⌡
then performed a u sub
u = x+2
du = 1dx
y = ⌠du/u
⌡
y = ln abs(u)+c
y = ln abs(x+2)+c
but the sanswer is supposed to be
y = ln abs ((x+2)/2) + c
Where did I mess up? and where did the 2 in the denominatior of the (x+2) come from?
Not sure where the 2 came from, but since
ln(u/2) = ln(u) - ln(2)
just fold the ln(2) into the C.
okay thank you!
i don't know answer.i am tring to find answer
To solve the differential equation, you started correctly by separating the variables:
dy = 1/(x+2) dx.
Then you integrated both sides with respect to their respective variables:
∫ dy = ∫ 1/(x+2) dx.
However, it seems you made a mistake during the integration. Let's correct it:
∫ dy = ∫ 1/(x+2) dx,
Integrating the left side gives:
y + C1 = ∫ dx.
The integral of dx is simply x, so we have:
y + C1 = x + C2.
Combining the constants of integration, we can write this as:
y = x + (C2 - C1).
Now let's address the discrepancy between your answer (ln |x+2| + C) and the expected answer (ln |(x+2)/2| + C).
In your solution, you considered the substitution u = x+2, which is correct. The differential du = dx. However, your mistake lies in the substitution of du without adjusting the limits of integration.
When you initially set up the integral:
∫ du/u,
you have to account for the change in variables and adjust the limits accordingly. The original limits of integration will change when substituting u = x+2.
The original limits for x were -2 to x. Substituting u = x+2, the new limits become u(-2) to u(x), which are 0 to (x+2).
Therefore, you need to rewrite the definite integral as follows:
∫[0 to (x+2)] du/u.
Now integrating this expression gives:
ln |u| + C3.
Substituting back u = x+2:
ln |x+2| + C3.
Let's denote C3 = C - ln|2|. Then we have:
ln |x+2| + C - ln|2| = ln |x+2| + C',
where C' = C - ln|2| is a new constant.
Hence, the correct solution is:
y = ln |x+2| + C',
which matches the expected answer: ln |(x+2)/2| + C.