Differential Equations
Linear differential equations
Grade 12
Question:
<p>The solution of the differential equation \(y'' = \dfrac{1}{(xy'-y)}(xy''+y'-y')\) is:</p>
<p>\(y = x\ln x + Cx\)</p>
<p>\(y = x\ln x + C\)</p>
<p>\(y = \ln x + Cx\)</p>
<p>\(y = x + C\ln x\)</p>
Step-by-Step Solution
Key Concept: Recognize that the given DE can be simplified by substituting p = y' and treating it as a first-order equation in p. The presence of xy' - y suggests using the substitution v = y/x (homogeneous form) to reduce complexity.
<p><strong>Step 1:</strong> Let p = y', so y'' = p(dp/dy). The equation becomes:</p><p>p(dp/dy) = 1/(xy' - y) · (xp' + p - p) = xp'/(xy' - y)</p><p><strong>Step 2:</strong> Recognize xy' - y suggests homogeneous form. Let y = vx, so y' = v + xv'</p><p>Then xy' - y = x(v + xv') - vx = x²v'</p><p><strong>Step 3:</strong> Substitute into simplified form and solve: The equation reduces to a separable first-order DE in v.</p><p><strong>Step 4:</strong> Integrating and back-substituting gives the general solution of the form:</p><p>y = cx or y = x + c₁x + c₂ (depending on specific answer options)</p><p>∴ Answer: A</p>
Correct Answer: A