Solve the differential equation dy/dx + y*cot(x) = 2*x + x2∗cot(x) given y(pi/2) = 0.
Get the complete, step-by-step math solution for: "Solve the differential equation dy/dx + y*cot(x) = 2*x + x²*cot(x) given y(pi/2) = 0.". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Identify the form of the differential equation
The given differential equation is of the form dxdy+ycot(x)=2x+x2cot(x). This is a first-order linear differential equation, which can be written in the standard form dxdy+P(x)y=Q(x). By comparing the given equation with the standard form, we can identify P(x) and Q(x).
Step 2: Identify P(x) and Q(x)
From the standard form, we can see that P(x) is the coefficient of y and Q(x) is the term on the right-hand side of the equation. In this case, P(x)=cot(x) and Q(x)=2x+x2cot(x).
Step 3: Calculate the integrating factor
The integrating factor (IF) for a linear differential equation is given by the formula e∫P(x)dx. We substitute P(x)=cot(x) into the formula and integrate. The integral of cot(x) is ln∣sin(x)∣. Assuming sin(x)>0 for its domain, the integrating factor simplifies to sin(x).
Step 4: Multiply the differential equation by the integrating factor
Now we multiply both sides of the original differential equation by the integrating factor, sin(x). This step transforms the left-hand side of the equation into the derivative of the product of y and the integrating factor, which is dxd(y⋅IF).
Step 5: Simplify the equation
The left-hand side of the equation becomes dxd(ysin(x)) because dxd(ysin(x))=dxdysin(x)+ycos(x). Since cot(x)sin(x)=sin(x)cos(x)sin(x)=cos(x), the equation simplifies to dxd(ysin(x))=2xsin(x)+x2cos(x). We can recognize the right-hand side as the derivative of x2sin(x) using the product rule: dxd(x2sin(x))=2xsin(x)+x2cos(x).
Step 6: Integrate both sides
We integrate both sides of the simplified equation with respect to x. The integral of dxd(ysin(x)) is simply ysin(x). On the right-hand side, we recognize (2xsin(x)+x2cos(x)) as the derivative of x2sin(x), so its integral is x2sin(x)+C, where C is the constant of integration.
Step 7: Solve for y
To find the general solution for y, we divide both sides of the equation by sin(x). This gives us y=sin(x)x2sin(x)+C, which simplifies to y=x2+Ccsc(x).
Step 8: Apply the initial condition
We are given the initial condition y(2π)=0. We substitute x=2π and y=0 into the general solution to find the value of the constant C. Since csc(2π)=1, we get 0=4π2+C, which means C=−4π2.
Step 9: Substitute C back into the general solution
Finally, we substitute the value of C back into the general solution for y. This gives us the particular solution to the differential equation that satisfies the given initial condition.