dy/dx=cos(7x)+5/pi
initial condition y(pi/4)=2
To find the function y(x) that satisfies the given differential equation dy/dx = cos(7x) + 5/π, and the initial condition y(pi/4) = 2, we can use the method of integration.
Step 1: Integrate both sides of the differential equation with respect to x. The left side integrates to y(x), and on the right side, we have to integrate cos(7x) dx:
∫ dy/dx dx = ∫ (cos(7x) + 5/π) dx
Step 2: On the left side, the integral of dy/dx dx is simply y(x). On the right side, we need to integrate cos(7x) and add the constant of integration C:
y(x) = ∫ cos(7x) dx + ∫ (5/π) dx + C
Step 3: Integrate each term separately:
∫ cos(7x) dx = (1/7) sin(7x) + K1,
∫ (5/π) dx = (5/π) x + K2,
where K1 and K2 are constants of integration.
Therefore, the equation becomes:
y(x) = (1/7) sin(7x) + (5/π) x + C,
where C = K1 + K2 is the constant of integration.
Step 4: Use the initial condition y(pi/4) = 2 to find the value of C:
2 = (1/7) sin(7(pi/4)) + (5/π) (pi/4) + C,
2 = (1/7) sin(7(pi/4)) + (5/4) + C.
Solve for C:
C = 2 - (1/7) sin(7(pi/4)) - (5/4).
Step 5: Substitute the value of C back into the equation:
y(x) = (1/7) sin(7x) + (5/π) x + 2 - (1/7) sin(7(pi/4)) - (5/4).
So, the function that satisfies the given differential equation dy/dx = cos(7x) + 5/π, and the initial condition y(pi/4) = 2, is:
y(x) = (1/7) sin(7x) + (5/π) x + 2 - (1/7) sin(7(pi/4)) - (5/4).