The general solution of the differential equa | vedclass

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(B) Given the differential equation: $e^{\frac{1}{2}\left(\frac{dy}{dx}\right)}=3^x$ Taking the natural logarithm on both sides: $\frac{1}{2}\frac{dy}

Vedclass · Accepted Answer

$y=x^2 \log 3+C$

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(B) Given the differential equation: $e^{\frac{1}{2}\left(\frac{dy}{dx}\right)}=3^x$ Taking the natural logarithm on both sides: $\frac{1}{2}\frac{dy}{dx} = \log_e(3^x)$ Using the property $\log(a^b) = b \log a$,we get: $\frac{1}{2}\frac{dy}{dx} = x \log_e 3$ Multiplying both sides by $2$: $\frac{dy}{dx} = 2x \log_e 3$ Integrating both sides with respect to $x$: $y = \int (2 \log_e 3) x \, dx$ $y = (2 \log_e 3) \frac{x^2}{2} + C$ $y = x^2 \log_e 3 + C$

The general solution of the differential equation $e^{\frac{1}{2}\left(\frac{dy}{dx}\right)}=3^x$ is (where $C$ is a constant of integration).

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