Question

If y=e^mtan⁻¹x,prove(1+x² )y₂+(2x-m)y₁=0

Answer 1

Answer:

$\bf\:(1+x^2)y_2+y_1(2x-m)=0$

Step-by-step explanation:

If y=e^mtan⁻¹x,prove(1+x² )y₂+(2x-m)y₁=0

$y=e^{m\:tan^{-1}x}$

Differentiate with respect to x

$y_1=e^{m\:tan^{-1}x}(m\frac{1}{1+x^2})$

$y_1=y(\frac{m}{1+x^2})$

$\implies\:(1+x^2)y_1=my$

differentiate with respect to x once again

$(1+x^2)y_2+y_1(2x)=my_1$

$(1+x^2)y_2+y_1(2x)-my_1=0$

$\implies\boxed{\bf\:(1+x^2)y_2+y_1(2x-m)=0}$