Question

The differential equation $\frac{dy}{dx}+4y=5$ is valied in the domain $0\le x\le 1$ with $y\left(0\right)=2.25$. The solution of the differential equation is

• A. $y=e^{4x}+5$
• B. $y=e^{-4x}+1.25$
• C. $y=e^{-4x}+5$
• D. $y=e^{4x}+1.25$

Answer 1

The correct option is B $y=e^{-4x}+1.25$

$\frac{dy}{dx}+4y=5$
$I.F=e^{\int 4.dx}=e^{4x}$
$\Rightarrow y\times I.F=\int 5\times (I.F)dx+C$
$\Rightarrow y.e^{4x}=5.\frac{e^{4x}}{4}+C$
$\Rightarrow 2.25\times 1=\frac{5.e^o}{4}+C$ (given $y\left(0\right)=2.25$)
$\Rightarrow C=1$
$y.e^{4x}=\frac{5}{4}{e}^{4x}+1$
$\Rightarrow y=\frac{5}{4}+e^{-4x}$
$\Rightarrow y=e^{-4x}+1.25$