<p>Find the differential equation of the family of curves \(y = e^x(A\cos x + B\sin x)\) where \(A, B\) are arbitrary constants. Also write its order and degree.</p>
Step-by-Step Solution
Key Concept: Eliminate two arbitrary constants A and B by differentiating twice and forming a system of equations. The resulting differential equation after elimination will have order equal to the number of constants and degree 1 (since no derivatives are raised to powers).
<p><strong>Step 1:</strong> Start with y = e<sup>x</sup>(A cos x + B sin x)</p><p><strong>Step 2:</strong> Differentiate once using product rule:<br/>dy/dx = e<sup>x</sup>(A cos x + B sin x) + e<sup>x</sup>(-A sin x + B cos x)<br/>dy/dx = y + e<sup>x</sup>(B cos x - A sin x)</p><p><strong>Step 3:</strong> Differentiate again:<br/>d²y/dx² = dy/dx + e<sup>x</sup>(B cos x - A sin x) + e<sup>x</sup>(-B sin x - A cos x)<br/>d²y/dx² = dy/dx + (dy/dx - y) + e<sup>x</sup>(-B sin x - A cos x)<br/>d²y/dx² = 2(dy/dx) - y - e<sup>x</sup>(A cos x + B sin x)<br/>d²y/dx² = 2(dy/dx) - y - y</p><p><strong>Step 4:</strong> Simplify to eliminate A and B:<br/>d²y/dx² - 2(dy/dx) + 2y = 0</p><p><strong>Step 5:</strong> Identify order and degree:<br/>Highest derivative is d²y/dx² → Order = 2<br/>Power of d²y/dx² is 1 → Degree = 1</p><p><strong>∴ Answer: Order 2, Degree 1</strong></p>
Correct Answer: Order 2, Degree 1