Final answer:
To solve the given differential equation, we need to find the integrating factor and use it to transform the equation. The integrating factor is obtained by taking the exponential of the integral of the coefficient of y. We then multiply the equation by the integrating factor and integrate both sides to obtain the solution.
Step-by-step explanation:
To solve the given differential equation xdy + (3x + 1) y dx = e⁻³ˣ dx, we need to find the integrating factor. The integrating factor is given by the exponential of the integral of the coefficient of y, which in this case is (3x + 1) dx.
Integrating (3x + 1) dx, we get (3/2)x² + x + C, where C is a constant of integration. Taking the exponential of this result, the integrating factor is given by e^((3/2)x² + x + C).
Multiplying the given differential equation by the integrating factor, we get the new equation: (x e^((3/2)x² + x + C)) dy + (3x^2 e^((3/2)x² + x + C) + x e^((3/2)x² + x + C)) y dx = e⁻³ˣ e^((3/2)x² + x + C) dx.
Simplifying this equation, we get d(y e^((3/2)x² + x + C)) = e⁻³ˣ e^((3/2)x² + x + C) dx.
Integrating both sides, we get y e^((3/2)x² + x + C) = ∫(e⁻³ˣ e^((3/2)x² + x + C)) dx.
Simplifying and solving for y, we get y = e^(-3x - (3/2)x² - x - C) (∫e⁻³ˣ e^((3/2)x² + x + C) dx + D), where D is a constant of integration.