<p>The solution of the differential equation \ \int\left(\dfrac{y+1}{y}\right)dy = \int e^x(\sin 2x - \cos^2 x)\,dx \ with \ x=0, y=1 \ gives the particular solution. Find the value of \ C:</p>
Step-by-Step Solution
Key Concept: Separate and integrate both sides carefully: the left side becomes y + ln|y| after integration, and the right side requires expanding sin 2x - cos²x and using integration by parts or substitution techniques. The constant C is determined using the initial condition x=0, y=1.
<p><strong>Step 1:</strong> Integrate the left side: ∫(y+1)/y dy = ∫(1 + 1/y)dy = y + ln|y| + C₁</p><p><strong>Step 2:</strong> Simplify the right side integrand: e^x(sin 2x - cos²x) = e^x(sin 2x - (1+cos 2x)/2) = e^x(sin 2x - 1/2 - cos 2x/2)</p><p><strong>Step 3:</strong> Integrate the right side using integration by parts for e^x terms. For ∫e^x sin 2x dx and ∫e^x cos 2x dx, use the formula ∫e^(ax)sin(bx)dx = e^(ax)(a·sin(bx) - b·cos(bx))/(a²+b²)</p><p><strong>Step 4:</strong> Right side = e^x(sin 2x/5 - 2cos 2x/5) - x/2 + C₂</p><p><strong>Step 5:</strong> Equate: y + ln|y| = e^x(sin 2x/5 - 2cos 2x/5) - x/2 + C</p><p><strong>Step 6:</strong> Apply initial condition x=0, y=1: 1 + ln(1) = e⁰(0 - 2/5) - 0 + C → 1 = -2/5 + C</p><p><strong>Step 7:</strong> Therefore, C = 1 + 2/5 = 7/5</p><p>∴ Answer: D (C = 7/5)</p>
Correct Answer: D