Find the general solution of the following differential equations :
`(dy)/(dx)=(4+x^(2))(9+y^(2))`

Answer

To find the general solution of the differential equation

\[ \frac{dy}{dx} = (4 + x^2)(9 + y^2), \]

we will proceed with the following steps:

### Step 1: Separate the variables
We can rewrite the equation to separate the variables \(y\) and \(x\):

\[ \frac{dy}{9 + y^2} = (4 + x^2)dx. \]

### Step 2: Integrate both sides
Next, we will integrate both sides. The left side requires the integral of \(\frac{1}{9 + y^2}\), which is a standard integral:

\[ \int \frac{dy}{9 + y^2} = \frac{1}{3} \tan^{-1}\left(\frac{y}{3}\right) + C_1, \]

where \(C_1\) is a constant of integration. The right side can be integrated as follows:

\[ \int (4 + x^2)dx = 4x + \frac{x^3}{3} + C_2, \]

where \(C_2\) is another constant of integration.

### Step 3: Combine the results
Now we can equate the results from both integrals:

\[ \frac{1}{3} \tan^{-1}\left(\frac{y}{3}\right) = 4x + \frac{x^3}{3} + C, \]

where \(C = C_2 - C_1\) is a new constant.

### Step 4: Solve for \(y\)
To express \(y\) in terms of \(x\), we can multiply both sides by 3:

\[ \tan^{-1}\left(\frac{y}{3}\right) = 12x + x^3 + 3C. \]

Now, take the tangent of both sides:

\[ \frac{y}{3} = \tan(12x + x^3 + 3C). \]

Finally, multiply by 3 to solve for \(y\):

\[ y = 3 \tan(12x + x^3 + 3C). \]

### General Solution
Thus, the general solution of the differential equation is:

\[ y = 3 \tan(12x + x^3 + C), \]

where \(C\) is a constant.

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