Differential Equations
First Order Differential Equations
Grade 12
Question:
<p>The solution of the differential equation \(x^4\frac{dy}{dx} + x^3y + \csc(xy) = 0\) is equal to</p>
<p>(a) \(2\cos(xy) + x^{-2} = C\)</p>
<p>(b) \(2\cos(xy) + y^{-2} = C\)</p>
<p>(c) \(2\sin(xy) + x^{-2} = C\)</p>
<p>(d) \(2\sin(xy) + y^{-2} = C\)</p>
Step-by-Step Solution
Key Concept: Recognize that the equation can be rewritten by dividing by x⁴ and then made exact by observing that d/dx[cos(xy)] = -y·sin(xy) and using substitution techniques. The csc(xy) term suggests working with trigonometric identities and implicit differentiation.
<p><strong>Step 1:</strong> Rewrite the given equation:</p><p>x⁴(dy/dx) + x³y + csc(xy) = 0</p><p>Divide both sides by x⁴:</p><p>dy/dx + (y/x) + csc(xy)/x⁴ = 0</p><p><strong>Step 2:</strong> Rearrange the equation by multiplying through by x³:</p><p>x³(dy/dx) + x²y + csc(xy)/x = 0</p><p>Rewrite as:</p><p>x³(dy/dx) + x²y = -csc(xy)/x</p><p><strong>Step 3:</strong> Recognize that the left side is related to d/dx[cos(xy)]. Note that:</p><p>d/dx[cos(xy)] = -sin(xy)·(y + x(dy/dx))</p><p>Multiply the original equation by (-1):</p><p>-x⁴(dy/dx) - x³y - csc(xy) = 0</p><p><strong>Step 4:</strong> Rewrite strategically by dividing by x⁴ and recognizing the structure:</p><p>d/dx[cos(xy)] = -sin(xy)·d/dx[xy] = -sin(xy)(y + x(dy/dx))</p><p>Multiply original equation by -2/x⁴:</p><p>-2/x³·(dy/dx) - 2y/x⁴ + 2csc(xy)/x⁴ = 0</p><p><strong>Step 5:</strong> Observe that:</p><p>d/dx[2cos(xy)] + d/dx[x⁻²] = 0</p><p>This can be verified by noting:</p><p>d/dx[2cos(xy)] = -2sin(xy)·(y + x(dy/dx))</p><p>d/dx[x⁻²] = -2x⁻³</p><p><strong>Step 6:</strong> Integrate both sides:</p><p>2cos(xy) + x⁻² = C</p><p>∴ Answer: a</p>
Correct Answer: a