solve the initial value problem by separation of variables dy/dx=y^(2)+1, y(1)=0
The integral of dy/(y^2 +1) equals the integral of dx. Integrate both sides from the y(1) = 0 point
arctany - arctan0 = (x - 1)
arctany = (x-1)
y = tan(x-1)
To solve the initial value problem by separation of variables, follow these steps:
Step 1: Rewrite the differential equation. Start by isolating the terms with "y" on one side of the equation:
dy/dx = y^2 + 1
Step 2: Separate the variables. Split the equation into the left-hand side and the right-hand side, each with their respective variables:
dy / (y^2 + 1) = dx
Step 3: Integrate both sides. Integrate both sides of the equation with respect to their respective variables:
∫dy / (y^2 + 1) = ∫dx
Step 4: Evaluate the left-hand side integral. The integral on the left-hand side ∫dy / (y^2 + 1) can be expressed as the inverse tangent function:
arctan(y) + C1 = x + C2
Note: C1 and C2 are constants of integration that appear during the process of integration.
Step 5: Simplify the equation. Combine the constants of integration by renaming their sum as a single constant:
arctan(y) = x + C
Step 6: Solve for y. Apply the inverse tangent to both sides of the equation to isolate y:
y = tan(x + C)
Step 7: Use the initial condition to find the specific solution. Substitute the initial condition y(1) = 0 into the equation to find the value of C:
0 = tan(1 + C)
Step 8: Find the value of C. To solve for C, take the inverse tangent of both sides and solve for C:
C = arctan(0) - 1 = -1
Step 9: Substitute the value of C into the general solution. Replace C in the general solution with its actual value:
y = tan(x - 1)
Thus, the solution to the initial value problem dy/dx = y^2 + 1, y(1) = 0 is y = tan(x - 1).