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A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
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
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
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