Differential Calculus

We are given:

%5Cfrac%7Bdy%7D%7Bdx%7D%20%3D%20-5%5Csin%20x%20-%203%5Ccos%20x.

Step 1: Compute the second derivative Differentiating

%5Cfrac%7Bdy%7D%7Bdx%7D

with respect to

x

, we get:

%5Cfrac%7Bd%5E2y%7D%7Bdx%5E2%7D%20%3D%20-5%5Ccos%20x%20%2B%203%5Csin%20x.

Step 2: Add

y

to the second derivative We know

y%20%3D%205%5Ccos%20x%20-%203%5Csin%20x

. Adding

y

%5Cfrac%7Bd%5E2y%7D%7Bdx%5E2%7D

, we have:

%5Cfrac%7Bd%5E2y%7D%7Bdx%5E2%7D%20%2B%20y%20%3D%20(-5%5Ccos%20x%20%2B%203%5Csin%20x)%20%2B%20(5%5Ccos%20x%20-%203%5Csin%20x).

Step 3: Simplify the expression Combine like terms:

%5Cfrac%7Bd%5E2y%7D%7Bdx%5E2%7D%20%2B%20y%20%3D%20-5%5Ccos%20x%20%2B%205%5Ccos%20x%20%2B%203%5Csin%20x%20-%203%5Csin%20x%20%3D%200.

Conclusion: The equation is satisfied, confirming that:

%5Cfrac%7Bd%5E2y%7D%7Bdx%5E2%7D%20%2B%20y%20%3D%200.

Chapter Questions
1 MCQs