Differential Equations
Bernoulli's Differential Equation
Grade 12
Question:
<p>Consider the differential equation<br/>\[ 2y\frac{dy}{dx} + y^2 \sec x = \tan x \]<br/>Which of the following is true about the method of solving this equation?</p>
<p>Substitute \( z = y^2 \) to reduce to linear ODE</p>
<p>It is a Bernoulli equation in \( y \)</p>
<p>Substitute \( z = y^3 \)</p>
<p>It is a linear ODE in \( y \)</p>
Step-by-Step Solution
Key Concept: Recognize that 2y(dy/dx) is the derivative of y² with respect to x, allowing a substitution v = y² to transform this into a linear first-order ODE in v.
<p><strong>Step 1:</strong> Recognize the pattern. Notice that 2y(dy/dx) = d(y²)/dx. Let v = y², then dv/dx = 2y(dy/dx).</p><p><strong>Step 2:</strong> Substitute into the original equation: dv/dx + v·sec x = tan x. This is now a linear first-order ODE of the form dv/dx + P(x)v = Q(x), where P(x) = sec x and Q(x) = tan x.</p><p><strong>Step 3:</strong> Find the integrating factor: μ(x) = e^(∫sec x dx) = e^(ln|sec x + tan x|) = sec x + tan x.</p><p><strong>Step 4:</strong> Multiply through by μ(x) and solve using the standard linear ODE method.</p><p><strong>Step 5:</strong> The method relies on recognizing the substitution that converts a quasi-linear equation into a standard linear form.</p><p>∴ <strong>Answer: A</strong> - The equation is solved using substitution v = y² to convert it into a linear first-order differential equation.</p>
Correct Answer: A