Find the solution of the differential equation that satisfies the given initial condition.
Passo 1
We have the ODE
with
. This is a first‐order linear ODE. To put it in standard form, divide both sides by
(nonzero for
) and use
, giving
Passo 2
Next we find the integrating factor. For an equation of the form
, the integrating factor is
. Here
, so
Passo 3
Multiply the standard‐form ODE by the integrating factor
. The left side becomes the derivative of
and the right side is
So we get
Passo 4
Integrate both sides with respect to
. On the left we get
. On the right use the substitution
,
:
Thus
Passo 5
Solve for
by multiplying by
:
This is the general solution.
Passo 6
Now impose the initial condition
. Since
, we have
Passo 7
Substitute
into the general solution to get the particular solution.