We use integrating factors to solve linear first-order differential equations. The integrating factor is typically labelled as either $\mu(x)$ or $I(x)$. Find the integrating factor $\mu(x)$ for the differential equation $\mu(x) = $ $\frac{dy}{dx} = 42x^5y + 4.$
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Step 1: Consider the differential equation in the form \[ \frac{dx}{dt} + P(t)x = Q(t) \] Show more…
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The differential equation is a linear differential equation: Convert the equation to standard form (use the prime notation for the derivative). The integrating factor is. After multiplying both sides by the integrating factor and undoing the product rule, we get the new differential equation: Integrating both sides, we get the algebraic equation. Solving for y, the solution to the differential equation is y. The differential equation is also a separable equation: Integrating both sides, we get the algebraic equation. Solving for y, we get y (use C for the constant in your answer).
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