Solve the differential equation dy/y=8x(4x^2-1)^(1/2)dx

 Exercise

$\frac{dy}{y}=8x\sqrt{4x^2-1}\:dx$

 

Integrate both sides of the differential equation, the left side with respect to $y$, and the right side with respect to $x$

$\int\frac{1}{y}dy=\int8x\sqrt{4x^2-1}dx$

 

Solve the integral $\int\frac{1}{y}dy$ and replace the result in the differential equation

$\ln\left|y\right|=\int8x\sqrt{4x^2-1}dx$

 

Solve the integral $\int8x\sqrt{4x^2-1}dx$ and replace the result in the differential equation

$\ln\left|y\right|=\frac{16\sqrt{\left(x^2-\frac{1}{4}\right)^{3}}}{3}+C_0$

 

Find the explicit solution to the differential equation. We need to isolate the variable $y$

$y=C_1e^{\frac{16\sqrt{\left(x^2-\frac{1}{4}\right)^{3}}}{3}}$

$y=C_1e^{\frac{16\sqrt{\left(x^2-\frac{1}{4}\right)^{3}}}{3}}$

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