(a) Find the consumer's surplus for the demand function $p = 25 - x - x^2$, where the prevailing market price $p_0 = 19$. OR (b) Solve the following initial value differential equation : $x \frac{dy}{dx} = y - x \tan(\frac{y}{x})$, given $y = \frac{\pi}{2}$ when $x = 1$.

Step 1: Find the quantity demanded at market price

To find the quantity demanded ($x_0$) at the prevailing market price ($p_0 = 19$), we set the demand function equal to the market price and solve for $x$. This gives us a quadratic equation.

Step 2: Solve the quadratic equation for x

Rearranging the equation $19 = 25 - x - x^2$ gives $x^2 + x - 6 = 0$. Factoring this quadratic equation yields $(x+3)(x-2) = 0$. Since quantity cannot be negative, we take the positive value, $x_0 = 2$.

Step 3: Apply the consumer's surplus formula

The consumer's surplus (CS) is calculated by integrating the difference between the demand function $p(x)$ and the market price $p_0$ from $0$ to the quantity demanded $x_0$.

Step 4: Substitute values and integrate

Substitute the demand function $p(x) = 25 - x - x^2$ and the market price $p_0 = 19$ into the formula. Simplify the integrand to $6 - x - x^2$ and then perform the integration.

Step 5: Evaluate the definite integral

Evaluate the definite integral by plugging in the upper limit $x=2$ and the lower limit $x=0$. Subtract the value at the lower limit from the value at the upper limit to find the consumer's surplus.

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