Final answer:
To find the general solution of the given differential equation, we first rewrite it in standard linear form and then use the integrating factor e^(3x) to multiply both sides. This transforms the equation so that the left side becomes the derivative of the product u(x)y, allowing us to integrate with respect to x and solve for y.
Step-by-step explanation:
To find the general solution to the given differential equation x²+3xy+x\frac{dy}{dx} = 0, we must first identify it as a first-order linear differential equation which can be written in standard form as \frac{dy}{dx} + P(x)y = Q(x), where P(x) and Q(x) are functions of x. In this case, by dividing by x (assuming x ≠ 0), we rewrite the equation as \frac{dy}{dx} + 3y = -x, which now appears in a more familiar linear form with P(x) being 3 and Q(x) being -x.
The integrating factor u(x) given as e^(3x) is used to multiply both sides of the equation to facilitate integration. Thus, the left side of the equation, upon multiplication by the integrating factor, becomes the derivative of the product u(x)y, which we can integrate with respect to x. Through integration, we obtain the value of u(x)y, and then we can solve for y, the general solution. It's important to consider the given integrating factor only when the problem requires it, and the equation matches the format where such an integrating factor is applicable.