Answer:
Given differential equation,
[tex]\frac{dy}{dx}+5y=7[/tex]
[tex]\frac{dy}{dx}=7-5y[/tex]
[tex]\implies \frac{dy}{7-5y}=dx[/tex]
Taking integration both sides,
[tex]\int \frac{dy}{7-5y}=\int dx[/tex]
Put 7 - 5y = u ⇒ -5 dy = du ⇒ dy = -du/5,
[tex]-\frac{1}{5} \int \frac{du}{u} = \log x + C[/tex]
[tex]-\frac{1}{5} \log u = \log x + C[/tex]
[tex]-\frac{1}{5}\log(7-5y) = \log x + C---(1)[/tex]
Here, x = 0, y = 0
[tex]\implies -\frac{1}{5} \log 7= C[/tex]
Hence, from equation (1),
[tex]-\frac{1}{5}\log(7-5y)=\log x -\frac{1}{5}log 7[/tex]
[tex]\log(7-5y)=\log (\frac{x}{7^\frac{1}{5}})[/tex]
[tex]7-5y=\frac{x}{7^\frac{1}{5}}[/tex]
[tex]7-\frac{x}{7^\frac{1}{5}}=5y[/tex]
[tex]\implies y=\frac{1}{5}(7-\frac{x}{7^\frac{1}{5}})[/tex]
