The solution of the equation xfrac{dy}{dx} + | vedclass

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(B) Given the linear differential equation: $x\frac{dy}{dx} + 3y = x$. Divide the entire equation by $x$ to get it into the standard form $\frac{dy}{d

Vedclass · Accepted Answer

$x^3y = \frac{x^4}{4} + c$

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(B) Given the linear differential equation: $x\frac{dy}{dx} + 3y = x$. Divide the entire equation by $x$ to get it into the standard form $\frac{dy}{dx} + Py = Q$: $\frac{dy}{dx} + \frac{3}{x}y = 1$. Here,$P = \frac{3}{x}$ and $Q = 1$. Calculate the Integrating Factor $(I.F.)$: $I.F. = e^{\int P dx} = e^{\int \frac{3}{x} dx} = e^{3 \ln|x|} = e^{\ln|x^3|} = x^3$. The general solution is given by $y \cdot (I.F.) = \int Q \cdot (I.F.) dx + c$. Substituting the values: $y \cdot x^3 = \int 1 \cdot x^3 dx + c$. $y x^3 = \frac{x^4}{4} + c$.

The solution of the equation $x\frac{dy}{dx} + 3y = x$ is

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