If cos x frac{dy}{dx} - y sin x = 6 x,0

Question

(D) Given the differential equation: $\cos x \frac{dy}{dx} - y \sin x = 6 x$. Divide the entire equation by $\cos x$: $\frac{dy}{dx} - y 	an x = 6 x

Vedclass · Accepted Answer

$y \cdot \cos x = 3 x^2 + c$,where $c$ is a constant of integration.

Vedclass · Answer

(D) Given the differential equation: $\cos x \frac{dy}{dx} - y \sin x = 6 x$. Divide the entire equation by $\cos x$: $\frac{dy}{dx} - y 	an x = 6 x \sec x$. This is a linear differential equation of the form $\frac{dy}{dx} + Py = Q$,where $P = -	an x$ and $Q = 6 x \sec x$. Calculate the Integrating Factor $(IF)$: $IF = e^{\int P dx} = e^{-\int 	an x dx} = e^{\ln(\cos x)} = \cos x$. The general solution is given by: $y \cdot (IF) = \int Q \cdot (IF) dx + c$. Substituting the values: $y \cdot \cos x = \int (6 x \sec x) \cdot \cos x dx + c$. Since $\sec x \cdot \cos x = 1$,we have: $y \cos x = \int 6 x dx + c$. Integrating $6x$ with respect to $x$: $y \cos x = 3 x^2 + c$.

If $\cos x \frac{dy}{dx} - y \sin x = 6 x$,$0 < x < \frac{\pi}{2}$,then the general solution of the differential equation is

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