Differential Equations
First Order Linear ODEs
Grade 12
Question:
<p>The solution of \(3x(1 - x^2)y^2\frac{dy}{dx} + (2x^2 - 1)y^3 = ax^3\) is</p>
<p>(A) \(y^3 = ax + c\sqrt{1 - x^2}\)</p>
<p>(B) \(y^3 = ax + cx^2\)</p>
<p>(C) \(y^2 = ax + c\sqrt{1 - x^2}\)</p>
<p>(D) None of these</p>
Step-by-Step Solution
Key Concept: Use substitution to convert to a linear first-order ODE and apply standard solution techniques.
<p>Rearrange and recognize this as a linear differential equation in $y^3$. Divide by $3x(1-x^2)y^2$ and use integrating factor method or substitution $u = y^3$ to solve.</p>
Correct Answer: A