What is the integrating factor?

Calculus Level 2

Consider the differential equation below,

$\frac{1}{2}dx-(y+\sqrt{y^{2}-x})dy=0$

Find the integrating factor, i.e., the term when multiply to the expression above make the equation exact.

The answer is of the form $(y^{b}-x^{c})^{-a}$ , where, $a$ is in its simplest fraction form (if it is a fraction), $b$ and $c$ are positive integers. Find $a+b+c$ .

Note that the arbitrary constant is not included in the integrating factor.

The answer is 3.5.

1 solution

Karan Chatrath
Feb 23, 2020

Consider the given equation:

$\frac{1}{2}dx = \left(y + \sqrt{y^2 - x}\right)dy$ $\implies dx - 2y \ dy = dy\sqrt{y^2 - x}$ Recognising that:

$dx - 2y \ dy = -d\left(y^2 - x\right)$

Simplifies the ODE to

$\implies -d\left(y^2 - x\right) = dy\sqrt{y^2 - x}$

Multiplying both sides by $\frac{1}{\sqrt{y^2 - x}}$ :

$\implies \frac{-d\left(y^2 - x\right)}{\sqrt{y^2 - x}} = dy$

The above equation is exact. Therefore the term required to be multiplied to the expression to make it exact is:

$\boxed{\frac{1}{\sqrt{y^2 - x}}}$

Therefore:

$\boxed{a = 0.5 \ ; \ b = 2 \ ; \ c = 1}$

What is the integrating factor?

The answer is 3.5.

1 solution

0 pending reports