To solve this differential equation, we will first rewrite it in standard form by dividing by (1 - x^2):
v''(x) + 2(1 + x^2)/(1 - x^2) * v'(x) = 0
Simplify the expression:
v''(x) + 2/(1 - x^2) * v'(x) = 0
v''(x) + 2/(1 - x^2) * v'(x) = 0
Now we will make the substitution u = v'(x) to rewrite the equation in terms of u:
u' + 2/(1 - x^2) * u = 0
u' = -2u/(1 - x^2)
Separate variables and integrate:
∫du/u = -2∫dx/(1 - x^2)
ln|u| = -2 arctan(x) + C
u = e^(-2 arctan(x) + C)
u = Ce^(-2 arctan(x))
Now substitute back v'(x) = Ce^(-2 arctan(x)):
v'(x) = Ce^(-2 arctan(x))
Integrate v'(x) to find v(x):
v(x) = ∫Ce^(-2 arctan(x)) dx
This integral may not have a simple closed-form solution, so the general solution to the differential equation is:
v(x) = Ce^(-2 arctan(x)) + D
where C and D are constants.
Solve the differential equation.
(1 - x^2)v''(x) + 2(1 + x^2)v'(x) = 0
1 answer