dy/dx=cos(7x)+5/pi
initial condition y(pi/4)=2
To find the particular solution to the given differential equation with the initial condition, you can follow these steps:
Step 1: Integrate both sides of the differential equation with respect to x to find the general solution:
∫ dy/dx dx = ∫ (cos(7x) + 5/π) dx
Integrating the left side:
∫ dy/dx dx = ∫ dy = y
Integrating the right side:
∫ (cos(7x) + 5/π) dx = ∫ cos(7x) dx + ∫ (5/π) dx
Using the integral of the cosine function:
∫ cos(7x) dx = (1/7) sin(7x) + C1,
where C1 is the constant of integration.
Using the integral of a constant:
∫ (5/π) dx = (5/π) x + C2,
where C2 is another constant of integration.
Therefore, the general solution to the differential equation is:
y = (1/7) sin(7x) + (5/π) x + C,
where C = C1 + C2 is the constant of integration.
Step 2: Use the given initial condition y(pi/4) = 2 to find the particular solution:
Substitute the value x = pi/4 and y = 2 into the general solution of y:
2 = (1/7) sin(7(pi/4)) + (5/π) (pi/4) + C
Simplifying the equation:
2 = (1/7) sin((7pi)/4) + 5/4 + C
Calculating sin((7pi)/4):
sin((7pi)/4) = sin(pi/4) = 1/sqrt(2)
Substituting back into the equation:
2 = (1/7)(1/sqrt(2)) + 5/4 + C
Simplifying further, you can combine the fractions:
2 = 1/(7sqrt(2)) + (35sqrt(2))/(28sqrt(2)) + C
Simplifying the fractions and simplifying sqrt(2):
2 = 1/(7sqrt(2)) + (35sqrt(2))/(4sqrt(2)) + C
2 = (1 + 35sqrt(2))/ (7sqrt(2)) + C
To find the value of C, rearrange the equation:
C = 2 - (1 + 35sqrt(2))/ (7sqrt(2))
Using a calculator or approximating the value, you'll find:
C ≈ 0.2037
Therefore, the particular solution to the differential equation with the initial condition is:
y = (1/7) sin(7x) + (5/π) x + 0.2037