Find the general solution of the given differential equation. x (dy/dx) + 3y = x³ - x

Solution:

Given,

Differential equation x (dy/dx) + 3y = x³ - x

Dividing both sides by x,

dy/dx + 3y/x = x² - 1 --- (1)

Rewriting the given equation in first order differential form 

(dy/dx) + Py = Q

Where, P = 3/x

Q = x² - 1

Now, integrating factor,

I.F. = e^{∫p dx}

I.F. = e^{∫(6/x) dx}

= x⁶

On multiplying equation (1) by I.F., we get,

I.F × y = ∫Q × I.F. dx

x⁶ × y = ∫(x² - 1)x⁶. dx

d(x⁶y) = ∫(x⁸ - x⁶).dx

d(x⁶y) = (x⁸ - x⁶) dx

∫d(x⁶y) = ∫(x⁸ - x⁶) dx

x⁶y = (1/9)x⁹ - (1/7)x⁷ + C

Dividing both sides by x⁶,

y = (1/9)x³ - (1/7)x + c/x⁶

Therefore, the general solution is y = (1/9)x³ - (1/7)x + c/x⁶.

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Find the general solution of the given differential equation. x (dy/dx) + 3y = x³ - x

Summary:

The general solution of the given differential equation x (dy/dx) + 3y = x³ - x is y = (1/9)x³ - (1/7)x + c/x⁶.