It is easy to check that for any value of c, the function
y=ce^–2x+e^–x
is solution of equation
y+2y=e^–x
Find the value of c for which the solution satisfies the initial condition y(–3)=4
can someone solve this one?
It is easy to check that for any value of c, the function
y=ce^–2x+e^–x
is solution of equation
y+2y=e^–x
Find the value of c for which the solution satisfies the initial condition y(–3)=4
can someone solve this one?
Are you sure you don't mean $\displaystyle y'+2y=e^{–x}$ ?
$\displaystyle y=ce^{-2x}+e^{-x}$
Using $\displaystyle y(-3)=4$
$\displaystyle 4=ce^{-2\times -3}+e^{-(-3)}$
$\displaystyle 4=ce^{6}+e^{3}$
$\displaystyle 4-e^{3}=ce^{6}$
$\displaystyle \frac{4-e^{3}}{e^{6}}=c$
$\displaystyle c=\frac{4-e^{3}}{e^{6}}$
Therefore
$\displaystyle y=\left(\frac{4-e^{3}}{e^{6}}\right)e^{–2x}+e^{–x}$