Question

Differential Equation of Variables
tan(y)dx-xlnxdy=0

Answer 1

Answer:

Step-by-step explanation:

We have,

$\rm{tan(y)\,dx-x\,\ln(x)\,dy=0}$

$\rm{\implies\,tan(y)\,dx=x\,\ln(x)\,dy}$

Separating variables, we get,

$\rm{\implies\,\dfrac{dx}{x\,\ln(x)}=\dfrac{dy}{tan(y)}}$

Integrating both sides,

$\displaystyle\rm{\implies\,\int\dfrac{dx}{x\,\ln(x)}=\int\dfrac{dy}{tan(y)}}$

Since, derivative of ln(x) is 1/x, so,

$\displaystyle\rm{\implies\,\int\dfrac{d(\ln(x))}{\ln(x)}=\int\,cot(y)\,dy}$

$\displaystyle\rm{\implies\,\ln\left|\ln(x)\right|=\ln\left|sin(y)\right|+\ln|C|}$

Where ln|C| is an arbitrary constant,

$\displaystyle\rm{\implies\,\ln\left|\ln(x)\right|=\ln\left|C\,sin(y)\right|}$

$\displaystyle\rm{\implies\,\ln(x)=C\,sin(y)}$