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$\frac{dy}{dt}+5y(t)=x(t)$

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Find derivatives using the quotient rule I

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Q.

\frac{dy}{dt}+5y(t)=x(t)

Find Integrating Factor: We are given a first-order linear non-homogeneous differential equation of the form $\frac{dy}{dt} + P(t)y = Q(t)$ , where $P(t) = 5$ and $Q(t) = x(t)$ . To solve this, we will first find the integrating factor, which is $e^{(\int P(t)dt)}$ . In this case, the integrating factor is $e^{(\int 5dt)} = e^{5t}$ .
Multiply by Integrating Factor: Now we multiply the entire differential equation by the integrating factor $e^{5t}$ to get $e^{5t} \cdot \frac{dy}{dt} + 5e^{5t} \cdot y(t) = e^{5t} \cdot x(t)$ .
Rewrite Differential Equation: Notice that the left side of the equation is the derivative of the product of the integrating factor and $y(t)$ , which is $\frac{d}{dt}(e^{5t} \cdot y(t))$ . So we can rewrite the equation as $\frac{d}{dt}(e^{5t} \cdot y(t)) = e^{5t} \cdot x(t)$ .
Integrate Both Sides: We will now integrate both sides of the equation with respect to $t$ . The left side becomes $e^{5t} \cdot y(t)$ , and the right side requires integrating $e^{5t} \cdot x(t)$ with respect to $t$ , which we will denote as $\int e^{5t} \cdot x(t) \, dt$ .
Solve for $y(t)$ : After integrating, we have $e^{5t} \cdot y(t) = \int e^{5t} \cdot x(t) \, dt + C$ , where $C$ is the constant of integration.
Solve for $y(t)$ : After integrating, we have $e^{5t} \cdot y(t) = \int e^{5t} \cdot x(t) \, dt + C$ , where $C$ is the constant of integration.To solve for $y(t)$ , we divide both sides by $e^{5t}$ , yielding $y(t) = \left(\frac{1}{e^{5t}}\right) \cdot \left(\int e^{5t} \cdot x(t) \, dt + C\right)$ .

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