Let \( y = y(x) \) be the solution of the differential equation \[ 2\cos x \frac{dy}{dx} = \sin 2x - 4y \sin x, \quad x \in \left( 0, \frac{\pi}{2} \right). \]
If \( y\left( \frac{\pi}{3} \right) = 0 \), then \( y\left( \frac{\pi}{4} \right) + y\left( \frac{\pi}{4} \right) \) is equal to ________.
Rewriting the equation:
\[ \frac{dy}{dx} = \frac{\sin 2x - 4y \sin x}{2 \cos x}. \]
This is a linear differential equation in the form:
\[ \frac{dy}{dx} + P(x) y = Q(x), \] where \( P(x) \) and \( Q(x) \) can be determined by comparing the given equation.
Solving this differential equation and applying the initial condition \( y\left( \frac{\pi}{3} \right) = 0 \), we find the value of \( y\left( \frac{\pi}{4} \right) \).
Final Answer: \( y\left( \frac{\pi}{4} \right) + y\left( \frac{\pi}{4} \right) = 4 \).
Total number of nucleophiles from the following is: \(\text{NH}_3, PhSH, (H_3C_2S)_2, H_2C = CH_2, OH−, H_3O+, (CH_3)_2CO, NCH_3\)
For the reaction, \[ H_2(g) + I_2(g) \rightleftharpoons 2HI(g) \] Attainment of equilibrium is predicted correctly by:
Match List - I with List - II:
List - I:
(A) \([ \text{MnBr}_4]^{2-}\)
(B) \([ \text{FeF}_6]^{3-}\)
(C) \([ \text{Co(C}_2\text{O}_4)_3]^{3-}\)
(D) \([ \text{Ni(CO)}_4]\)
List - II:
(I) d²sp³ diamagnetic
(II) sp²d² paramagnetic
(III) sp³ diamagnetic
(IV) sp³ paramagnetic