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Finding c1 and c2 to Satisfy Given Conditions

In this article, we will explore the process of finding the values of c1 and c2 so that the function y(x)=c1sinx+c2cosx satisfies the given conditions.

Condition 1: y(0)=0, y'(π/2)=1

To satisfy the first condition, we need to find the values of c1 and c2 that make y(0)=0 and y'(π/2)=1.

1. Evaluate y(0): y(0) = c1sin(0) + c2cos(0) y(0) = c2

   Since we want y(0)=0, we set c2=0.

2. Evaluate y'(x): y'(x) = c1cos(x) - c2sin(x)

3. Evaluate y'(π/2): y'(π/2) = c1cos(π/2) - c2sin(π/2) y'(π/2) = -c1

   Since we want y'(π/2)=1, we set c1=-1.

Therefore, the values of c1 and c2 that satisfy the first condition are: c1 = -1 c2 = 0

Condition 2: y(0)=1, y'(π)=1

To satisfy the second condition, we need to find the values of c1 and c2 that make y(0)=1 and y'(π)=1.

1. Evaluate y(0): y(0) = c1sin(0) + c2cos(0) y(0) = c2

   Since we want y(0)=1, we set c2=1.

2. Evaluate y'(x): y'(x) = c1cos(x) - c2sin(x)

3. Evaluate y'(π): y'(π) = c1cos(π) - c2sin(π) y'(π) = -c1 + 0 y'(π) = -c1

   Since we want y'(π)=1, we set c1=-1.

Therefore, the values of c1 and c2 that satisfy the second condition are: c1 = -1 c2 = 1

In conclusion, the function y(x)=c1sinx+c2cosx will satisfy the given conditions:

1. y(0)=0, y'(π/2)=1 when c1=-1 and c2=0.
2. y(0)=1, y'(π)=1 when c1=-1 and c2=1.