Question

For the differential equation

$y\left(8x-9y\right)dx+2x\left(x-3y\right)dy=0,$

which one of the following statements is TRUE?

1. The differential equation is not exact and has $x^2$ as an integrating factor
2. The differential equation is exact and homogeneous
3. The differential equation is not exact and does not have $x^2$ as an integrating factor
4. The differential equation is not homogeneous and has $x^2$ as an integrating factor

Answer: The differential equation is not exact and has $x^2$ as an integrating factor

Solution :-

We have $y\left(8x-9y\right)dx+2x\left(x-3y\right)dy=0$ . Let’s first check given differential equation is homogeneous or not.
For checking homogeneous replace $x$ by $\lambda x$ and $y$ by $\lambda y$ in functions only, we get

we got the original differential equation in last, so the given differential equation is homogeneous.

Now, let’s check given differential equation is exact or not.

In differential equation we have, $M=y\left(8x-9y\right)=8xy-9y^2$ and $N=2x\left(x-3y\right)=2x^2-6xy\mathrm{.}$

Here, $\frac{\partial M}{\partial y}\ne \frac{\partial N}{\partial x}$ . Hence, given differential equation is not exact.

Now, let’s check $x^2$ is an integrating factor or not.

We have,

Now, on comparing with $x^a{y}^b\left(mydx+nxdy\right)+x^{a^{\prime }}{y}^{b^{\prime }}\left(m^{\prime }ydx+n^{\prime }xdy\right)=0.$ We get,

Now,

from $eq\left(1\right)\times 2-eq\left(2\right)$ , we get $k=0\text{and}h=2$ .

Hence, Integrating Factor( $\text{I.F.}$ ) $=x^h{y}^k=x^2{y}^0=x^2\mathrm{.}$

So, the correct answer is option-1, $\to$ The differential equation is not exact and has $x^2$ as an integrating factor