Question

Solve the differential equation 0y%5Csqrt%7B3x%5E%7B2%7D%20-%201%20%7D" id="TexFormula1" title="\frac{1}{6x} * \frac{dy}{dx} = y\sqrt{3x^{2} - 1 }" alt="\frac{1}{6x} * \frac{dy}{dx} = y\sqrt{3x^{2} - 1 }" align="absmiddle" class="latex-formula">
A. $e^{2/3} \sqrt{(3x^{2} -1)^{3}}$
B. $C*e^{2/3} \sqrt{(3x^{2} -1)^{3}}$
C. None of these

Answer 1

This ODE is separable as

1/(6x) dy/dx = y √(3x ² - 1)

→   dy / y = 6x √(3x ² - 1) dx

Integrate both sides:

∫ dy / y = ∫ 6x √(3x ² - 1) dx

The left side is trivial. For the right side, substitute u = 3x ² - 1 and du = 6x dx :

∫ dy / y = ∫ √u du

ln|y | = u ³′² + C

(that is, u is raised to the 3/2 power)

ln|y | = (3x ² - 1)³′² + C

Solve for y by taking the exponential of both sides:

exp(ln|y |) = exp((3x ² - 1)³′² + C )

y = exp((3x ² - 1)³′²) × exp(C )

y = C exp((3x ² - 1)³′²)

which can be written as

y = C exp(√((3x ² - 1)³))

which makes the answer none of these; this solution can't be expressed as either option given in A or B.