The general solution of the differential equa | vedclass

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(B) Let $v = 9x + y + 5$. Then $\frac{dv}{dx} = 9 + \frac{dy}{dx}$, so $\frac{dy}{dx} = \frac{dv}{dx} - 9$.Substituting into the equation: $\frac{dv}{

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$\frac{1}{3} 	an^{-1} (\frac{9x + y + 5}{3}) = x + c$

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(B) Let $v = 9x + y + 5$. Then $\frac{dv}{dx} = 9 + \frac{dy}{dx}$, so $\frac{dy}{dx} = \frac{dv}{dx} - 9$.Substituting into the equation: $\frac{dv}{dx} - 9 = v^2 \implies \frac{dv}{dx} = v^2 + 9$.Separating variables: $\int \frac{dv}{v^2 + 3^2} = \int dx$.Using the integral formula $\int \frac{dx}{x^2 + a^2} = \frac{1}{a} 	an^{-1}(\frac{x}{a}) + c$, we get $\frac{1}{3} 	an^{-1}(\frac{v}{3}) = x + c$.Substituting $v = 9x + y + 5$ back: $\frac{1}{3} 	an^{-1}(\frac{9x + y + 5}{3}) = x + c$.

The general solution of the differential equation $\frac{dy}{dx} = (9x + y + 5)^2$ is...

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