For the exact differential equation,

$\frac{du}{dx}=\frac{-x{u}^2}{2+x^{2}u}$

which one of the following is the solution?

The correct answer is:
$\frac{1}{2}{x}^2{u}^2+2u$ = constant

Step-by-Step Explanation:

Step 1: Rearrange the given differential equation into differential form
From the provided image, the given differential equation is:
$$\frac{du}{dx}=\frac{-x{u}^2}{2+x^{2}u}$$
We can cross-multiply to separate the variables and differentials:
$$(2+x^{2}u)du=-x{u}^{2}dx$$
Rearranging all terms to one side gives:
$$(2+x^{2}u)du+x{u}^{2}dx=0$$

Step 2: Identify the components and verify the exactness condition
An ordinary differential equation of the form:
$$Mdu+Ndx=0$$
is exact if the partial derivatives satisfy:
$$\frac{\partial M}{\partial x}=\frac{\partial N}{\partial u}$$
(represented in the image using the symbol δ for partial derivatives: $\frac{\delta M}{\delta x}=\frac{\delta N}{\delta u}$).
By comparing the terms, we identify:
$$M=2+x^{2}u$$
$$N=x{u}^2$$
Now we calculate the partial derivatives:
• Differentiating M partially with respect to x (treating u as a constant):
$$\frac{\partial M}{\partial x}=0+2xu=2xu$$
• Differentiating N partially with respect to u (treating x as a constant):
$$\frac{\partial N}{\partial u}=2xu$$
Since $\frac{\partial M}{\partial x}=\frac{\partial N}{\partial u}=2xu$, the exactness condition is satisfied.

Step 3: Solve the exact differential equation
The general solution to the exact differential equation is given by:
$$\int Mdu+\int (\text{terms of }N\text{ not containing }u)dx=\text{Constant}$$
Here, we integrate M with respect to u (treating x as a constant):
$$\int (2+x^{2}u)du=2u+\frac{x^2{u}^2}{2}$$
Next, we identify terms in $N=x{u}^2$ that do not contain the variable u. Since the only term in N has u², there are no terms independent of u. Thus:
$$\int 0dx=0$$
Combining these integrals yields the solution:
$$2u+\frac{x^2{u}^2}{2}=\text{constant}$$
Which is algebraically equivalent to:
$$\frac{1}{2}{x}^2{u}^2+2u=\text{constant}$$