Differential Equations
Linear Differential Equations
Grade 12
Question:
<p>The solution of the differential equation \ (1+\tan y)\dfrac{dx}{dy} + 2x = (1+\tan y) \ is:</p>
<p>A) \ x e^y (\cos y + \sin y) = e^y \cos y + C</p>
<p>B) \ x e^y (\cos y + \sin y) = e^y \sin y + C</p>
<p>C) \ x e^y (\cos y - \sin y) = e^y \sin y + C</p>
<p>D) \ x e^y (\cos y + \sin y) = e^y \tan y + C</p>
Step-by-Step Solution
Key Concept: Recognize this as a linear differential equation in x as a function of y. Rearrange to standard form dx/dy + P(y)x = Q(y), then apply the integrating factor method with factor e^∫P(y)dy.
<p><strong>Step 1:</strong> Rearrange the equation into standard linear form.</p><p>Divide throughout by (1+tan y):</p><p>$$\frac{dx}{dy} + \frac{2x}{1+\tan y} = 1$$</p><p><strong>Step 2:</strong> Identify P(y) = 2/(1+tan y) and find integrating factor.</p><p>$$\text{I.F.} = e^{\int \frac{2}{1+\tan y}dy}$$</p><p><strong>Step 3:</strong> Simplify the integral using 1 + tan y = (cos y + sin y)/cos y.</p><p>$$\int \frac{2}{1+\tan y}dy = \int \frac{2\cos y}{\cos y + \sin y}dy = \ln(\cos y + \sin y)^2$$</p><p><strong>Step 4:</strong> Therefore I.F. = (cos y + sin y)²</p><p><strong>Step 5:</strong> Multiply equation by I.F. and integrate:</p><p>$$\frac{d}{dy}[x(\cos y + \sin y)^2] = (\cos y + \sin y)^2$$</p><p><strong>Step 6:</strong> Integrate both sides with respect to y using substitution u = cos y + sin y:</p><p>$$x(\cos y + \sin y)^2 = \frac{(\cos y + \sin y)^3}{3} + C$$</p><p>∴ Answer: B</p>
Correct Answer: B