What is the equation of the curve passing through the origin and satisfying the differential equation dy = (y tan x + sec x) dx ?

Options

A. y = x cos x
B. y cos x = x
C. xy = cos x
D. y sin = x

Answer

To solve the differential equation \( dy = (y \tan x + \sec x) dx \) and find the equation of the curve passing through the origin, we can follow these steps:

### Step 1: Rewrite the Differential Equation
We start with the given differential equation:
\[ dy = (y \tan x + \sec x) dx \]
We can rewrite this in the standard form:
\[ \frac{dy}{dx} = y \tan x + \sec x \]

### Step 2: Identify the Integrating Factor
The equation is of the form \( \frac{dy}{dx} - P(x)y = Q(x) \), where \( P(x) = -\tan x \) and \( Q(x) = \sec x \). We need to find the integrating factor \( \mu(x) \):
\[ \mu(x) = e^{\int -\tan x \, dx} \]
The integral of \( -\tan x \) is:
\[ -\int \tan x \, dx = -\log |\sec x| = \log |\cos x| \]
Thus, the integrating factor is:
\[ \mu(x) = e^{\log |\cos x|} = \cos x \]

### Step 3: Multiply the Equation by the Integrating Factor
Now we multiply the entire differential equation by the integrating factor \( \cos x \):
\[ \cos x \frac{dy}{dx} - y \sin x = 1 \]

### Step 4: Rearrange the Equation
This can be rearranged as:
\[ \cos x \frac{dy}{dx} - y \sin x = 1 \]

### Step 5: Integrate Both Sides
Next, we integrate both sides with respect to \( x \):
\[ \int \left( \cos x \frac{dy}{dx} - y \sin x \right) dx = \int 1 \, dx \]
The left side can be integrated using the product rule:
\[ \int d(y \cos x) = \int 1 \, dx \]
This gives us:
\[ y \cos x = x + C \]

### Step 6: Solve for \( y \)
Now, we solve for \( y \):
\[ y = \frac{x + C}{\cos x} \]

### Step 7: Apply the Initial Condition
Since the curve passes through the origin (0,0), we substitute \( x = 0 \) and \( y = 0 \):
\[ 0 = \frac{0 + C}{\cos(0)} \implies 0 = C \]
Thus, \( C = 0 \).

### Final Equation of the Curve
The equation of the curve is:
\[ y = \frac{x}{\cos x} \]