To find the general solution of the given differential equation
x dy/dx = 6y + x3 – x, we start by rewriting the equation in a more manageable form.
First, let’s rearrange the equation:
dy/dx = (6y + x3 – x)/x
This simplifies to:
dy/dx = 6(y/x) + x2 – 1
Now, we can use a substitution to facilitate solving the equation. Let’s define:
v = y/x so that y = vx.
Using the product rule, the derivative of y with respect to x can be expressed as:
dy/dx = v + x(dv/dx)
Substituting back into our original equation gives us:
v + x(dv/dx) = 6v + x2 – 1
Rearranging this equation results in:
x(dv/dx) = 5v + x2 – 1
This simplifies to:
dv/dx = (5v + x2 – 1)/x
Now we have a first-order linear ordinary differential equation in standard form:
dv/dx – (5/x)v = (x2 – 1)/x
Next, we find the integrating factor, μ(x), which is:
μ(x) = e-5ln|x| = x-5
Multiplying both sides of our equation by the integrating factor:
x-5dv/dx – 5x-6v = (x-3 – x-5)
This can be expressed as:
d/dx(v*x-5) = x-3 – x-5
Integrating both sides gives:
v*x-5 = -1/2 * x-2 + 1/4*x-4 + C
Where C is the constant of integration. Now solving for v:
v = (-1/2 * x3 + 1/4*x1 + C*x5)
Substituting back for y gives:
y = v*x = x*(-1/2 * x3 + 1/4*x1 + C*x5)
The general solution of the differential equation is:
y = -1/2 * x4 + 1/4 * x2 + C*x6
In conclusion, by transforming the original differential equation and utilizing substitution along with integrating factors, we successfully arrived at a general solution. This approach not only simplifies the equation but also leverages standard methods for solving first-order linear ordinary differential equations.