The diff. equation whose solutions are `y=x sin (x+A)`, A being constant is .........

Answer

To find the differential equation whose solutions are given by \( y = x \sin(x + A) \), where \( A \) is a constant, we can follow these steps:

### Step 1: Differentiate the given function
We start with the function:
\[ y = x \sin(x + A) \]
We will differentiate this with respect to \( x \). Using the product rule, where \( u = x \) and \( v = \sin(x + A) \):
\[ \frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx} \]
Calculating \( \frac{du}{dx} \) and \( \frac{dv}{dx} \):
- \( \frac{du}{dx} = 1 \)
- \( \frac{dv}{dx} = \cos(x + A) \)

Thus, we have:
\[ \frac{dy}{dx} = x \cos(x + A) + \sin(x + A) \]

### Step 2: Substitute \( \sin(x + A) \)
From the original equation, we know:
\[ \sin(x + A) = \frac{y}{x} \]
Substituting this into our derivative:
\[ \frac{dy}{dx} = x \cos(x + A) + \frac{y}{x} \]

### Step 3: Express \( \cos(x + A) \)
We can express \( \cos(x + A) \) using the identity:
\[ \cos(x + A) = \sqrt{1 - \sin^2(x + A)} \]
Substituting \( \sin(x + A) \):
\[ \cos(x + A) = \sqrt{1 - \left(\frac{y}{x}\right)^2} \]

### Step 4: Substitute back into the derivative
Now, substituting \( \cos(x + A) \) back into the derivative:
\[ \frac{dy}{dx} = x \sqrt{1 - \left(\frac{y}{x}\right)^2} + \frac{y}{x} \]

### Step 5: Clear the fractions
Multiply through by \( x \) to eliminate the fraction:
\[ x \frac{dy}{dx} = x^2 \sqrt{1 - \left(\frac{y}{x}\right)^2} + y \]

### Step 6: Rearranging the equation
Rearranging gives us:
\[ x \frac{dy}{dx} - y = x^2 \sqrt{1 - \left(\frac{y}{x}\right)^2} \]

### Step 7: Square both sides
To eliminate the square root, we square both sides:
\[ \left(x \frac{dy}{dx} - y\right)^2 = x^4 \left(1 - \left(\frac{y}{x}\right)^2\right) \]

### Step 8: Simplifying the equation
Expanding both sides leads to:
\[ \left(x \frac{dy}{dx} - y\right)^2 = x^4 - y^2 \]

### Final Step: Rearranging to standard form
Rearranging gives us the final form of the differential equation:
\[ x \frac{dy}{dx} - y^2 + x^2 y^2 = x^4 \]

Thus, the differential equation whose solutions are \( y = x \sin(x + A) \) is:
\[ x \frac{dy}{dx} - y^2 + x^2 y^2 = x^4 \]