Find the general solution of the given differential equation. x (dy/dx) + 4y = x³ - x

Solution:

Given,

Differential equation x (dy/dx) + 4y = x³ - x

Dividing both sides by x,

dy/dx + 4y/x = x² - 1 --- (1)

Rewriting the given equation in first order differential form

(dy/dx) + Py = Q

Where, P = 4/x

Q = x² - 1

Now, integrating factor,

I.F. = e^{∫p dx}

I.F. = e^∫(4/x) dx

= x⁴

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

I.F × y = ∫Q × I.F. dx

x⁴ × y = ∫(x² - 1)x⁴. Dx

x⁴(dy/dx) + 4x³y = x⁶−x⁴

d(x⁴y)/ dx = x⁶−x⁴

d(x⁴y) = (x⁶−x⁴) dx

∫d(x⁴y) = ∫(x⁶−x⁴) dx

x⁴y = (1/7)x⁷−(1/5)x⁵ + C

Dividing both sides by x⁴,

y = (1/7)x³−(1/5)x + C/x⁴

Therefore, the general solution is y = (1/7)x³−(1/5)x + C/x⁴

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Find the general solution of the given differential equation. x (dy/dx) + 4y = x³ - x

Summary:

The general solution of the given differential equation x (dy/dx) + 4y = x³ - x is y = (1/7)x³−(1/5)x + C/x⁴