the general solution of the given differential equation is:
y = (1/|x|^2) [(1/5)x^5 - (1/3)x^3 + C]
where C is the constant of integration.
To find the general solution of the given differential equation, we need to solve for y in terms of x. The differential equation is:
What is Integrating factor?
x dy/dx + 2y = x^3 - x
To solve this, we can use an integrating factor. First, we rearrange the equation in the standard form:
dy/dx + (2/x) y = (x^3 - x)/x
The integrating factor (IF) is defined as the exponential of the integral of the coefficient of y. In this case, the coefficient is (2/x), so the IF is:
IF = exp(∫(2/x) dx)
= exp(2 ln|x|)
= exp(ln|x|^2)
= |x|^2
Now, we multiply both sides of the differential equation by the integrating factor:
|x|^2(dy/dx) + (2|x|^2 / x) y = (x^3 - x)|x|^2 / x
Simplifying this expression, we have:
|x|^2(dy/dx) + 2|x|y = (x^3 - x)|x|
Now, we can rewrite the left-hand side as the derivative of (|x|^2y) with respect to x:
d/dx (|x|^2y) = (x^3 - x)|x|
Integrating both sides with respect to x, we get:
∫ d/dx (|x|^2y) dx = ∫ (x^3 - x)|x| dx
|x|^2y = ∫ (x^4 - x^2) dx
Integrating further, we have:
|x|^2y = (1/5)x^5 - (1/3)x^3 + C
Finally, we can solve for y:
y = (1/|x|^2) [(1/5)x^5 - (1/3)x^3 + C]
Therefore, the general solution of the given differential equation is:
y = (1/|x|^2) [(1/5)x^5 - (1/3)x^3 + C]
where C is the constant of integration.
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Find the inverse Laplace transforms of the following functions. First, perform partial-fraction expansion on G(s); then, use the Laplace transform table. (a). G(s)= 1 / s(s+2)(s+3) (b). G(s)= 10 / (s +1)^2(s+3) (c). G(s)= [100(s+2) / s(s^2 + 4)(s+1)] e^-x
(d). G(s)= 2(s+1) / s(s^2+s+2) (e). G(s)= 1 / (s+1)^3 (f). G(s)= 2(s^2+s+1) / s(s+1.5)(s^2 +5s+5)
(g). G(s)= [2+2se^(-x) + 4e^(-2x)] / [s^2 + 3s + 2] (h). G(s) = 2s+1 / (s^2 + 6s^2 +11s +6)
(i). G(s) = (3s^3 + 10s^2 + 8s + 5) / (s^4 + 5s^3 + 7s^2 + 5s +6)