Let $ y(x) $ be the solution of the differential equation $$ x^2 \frac{dy}{dx} + xy = x^2 + y^2, \quad x > \frac{1}{e}, $$ satisfying $ y(1) = 0 $. Then the value of $ 2 \cdot \frac{(y(e))^2}{y(e^2)} $ is ________.
The following table shows the ages of the patients admitted in a hospital during a year. Find the mode and the median of these data.\[\begin{array}{|c|c|c|c|c|c|c|} \hline Age (in years) & 5-15 & 15-25 & 25-35 & 35-45 & 45-55 & 55-65 \\ \hline \text{Number of patients} & \text{6} & \text{11} & \text{21} & \text{23} & \text{14} & \text{5} \\ \hline \end{array}\]
In the following figure \(\triangle\) ABC, B-D-C and BD = 7, BC = 20, then find \(\frac{A(\triangle ABD)}{A(\triangle ABC)}\).
The radius of a circle with centre 'P' is 10 cm. If chord AB of the circle subtends a right angle at P, find area of minor sector by using the following activity. (\(\pi = 3.14\))
Activity : r = 10 cm, \(\theta\) = 90\(^\circ\), \(\pi\) = 3.14. A(P-AXB) = \(\frac{\theta}{360} \times \boxed{\phantom{\pi r^2}}\) = \(\frac{\boxed{\phantom{90}}}{360} \times 3.14 \times 10^2\) = \(\frac{1}{4} \times \boxed{\phantom{314}}\) <br>A(P-AXB) = \(\boxed{\phantom{78.5}}\) sq. cm.
Find the Derivative \( \frac{dy}{dx} \)Given:\[ y = \cos(x^2) + \cos(2x) + \cos^2(x^2) + \cos(x^x) \]
