Find the general solution of the given differential equation. x(dy/dx) + 4y = x3 - x?
We will be using the First Order differential equation to solve this.
Answer: The general solution of the given differential equation x(dy/dx) + 4y = x3 - x is (1/7)X3- (1/5)x + (c /x4)
Let's solve this question step by step.
Differential equation. x(dy/dx) + 4y = x3 − x
Dividing both sides by x,
(dy/dx) + 4y/x = x2 − 1....(eq 1)
To solve the equation (dy/dx) + 4y/x = x2 − 1 we need to rewrite the given differential equation in First Order DE form :
(dy/dx) + Py = Q
P= 4/x, Q = x2 − 1
Now, integrating factor, I.F.=e∫ p dx
I.F.=e∫ (4/x) dx = x4
On multiplying (eq 1) by this 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
y = (1/7)x3−(1/5)x + C/x4