Solve dy/dx + y*tan(x) = sec(x) given y(0) = 1.
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Step-by-Step Solution
Step 1: Identify the form of the differential equation
The given differential equation dxdy + y tan(x)=sec(x) is a first-order linear differential equation. We can compare it to the standard form dxdy + P(x) y=Q(x) to identify P(x) and Q(x)$.
Step 2: Calculate the integrating factor
To solve a first-order linear differential equation, we need to find the integrating factor (IF). The integrating factor is given by the formula e∫P(x)dx In this case, P (x)=tan(x), and the integral of tan(x) is log∣sec(x)∣ Therefore, the integrating factor simplifies to sec(x).
Step 3: Apply the solution formula
The general solution to a first-order linear differential equation is given by y ⋅IF=∫Q(x)⋅IF \, dx .WesubstituteQ(x)=sec(x) and the integrating factor sec(x) into this formula. The integral of sec2(x)istan(x), so we get y sec(x)=tan(x)+C, where C is the constant of integration.
Step 4: Solve for y
To find the explicit solution for y we divide both sides of the equation by sec(x). We can rewrite tan(x) as cos(x)sin(x) and sec(x) as cos(x)1 After simplification, we obtain y =sin(x)+Ccos(x).
Step 5: Apply the initial condition
We are given the initial condition y(0) = 1 .Wesubstitutex=0andy=1 into the general solution y =sin(x)+Ccos(x). Since sin(0)=0 and cos(0)=1 we find that 1=0+C(1), which means C=1.
Step 6: Write the particular solution
Now that we have found the value of the constant C=1 we substitute it back into the general solution to get the particular solution for the given initial condition. The particular solution is y =sin(x)+cos(x).