Final answer:
To find the general solution to the given differential equation, we put it in standard form and find the integrating factor. We then separate the equation into two parts and solve them separately.
Step-by-step explanation:
To find the general solution to the given differential equation, we can put it in standard form by rearranging the terms:
x² - 3xy + x dx/dy = 0
Next, we need to find the integrating factor, which is denoted by rho(x). In this case, rho(x) = exp(integral(-3x dx)) = exp(-3x²/2).
Multiplying the entire equation by rho(x), we get:
(x² - 3xy + x dx/dy) * exp(-3x²/2) = 0
This can be simplified to:
(x²*exp(-3x²/2)) dx + (-3xy*exp(-3x²/2) + x dx/dy * exp(-3x²/2)) dy = 0
Now, let y(x) = u(x) * exp(-3x²/2), where u(x) is an unknown function of x. Differentiating with respect to x:
dy/dx = u'(x)*exp(-3x²/2) - 3x*u(x)*exp(-3x²/2) + u(x)*exp(-3x²/2)*d(u(x))/dx = u'(x)*exp(-3x²/2) - 3x*y(x) + u(x)*exp(-3x²/2)*d(u(x))/dx
Substituting the above expression into the original equation, we obtain:
(x²*exp(-3x²/2)) dx + (-3xy*exp(-3x²/2) + x(u'(x)*exp(-3x²/2) - 3x*y(x) + u(x)*exp(-3x²/2)*d(u(x))/dx)) dy = 0
Expanding and collecting the terms, we get:
(x²*exp(-3x²/2) + (-3xy*exp(-3x²/2)) dy + (x*u'(x) - 3x²*y(x) + u(x)*d(u(x))/dx) exp(-3x²/2) dy = 0
Since the equation is equal to 0, the coefficients of dx and dy must separately equal 0:
x²*exp(-3x²/2) + (-3xy*exp(-3x²/2)) = 0
x*u'(x) - 3x²*y(x) + u(x)*d(u(x))/dx = 0
To find the general solution, we solve each equation separately. The first equation is a separable differential equation that can be solved using standard techniques. The second equation is a linear first-order homogeneous differential equation that can be solved using an integrating factor.
By solving these equations, we can find the general solution to the given differential equation.