Differential Equations
First order differential equations
Grade 12
Question:
<p>Let \(f(x) \cdot g(y) + f'(x) \cdot g(y) = g'(y)\) with \(f(0) = 6\), \(g(0) = 1\). Which of the following are correct?</p>
<p>(a) \(f(x) = 1 + 5e^{-x}\)</p>
<p>(b) \(g(y) = e^y\)</p>
<p>(c) \(f(1) = 1 + \frac{5}{e}\)</p>
<p>(d) \(g(1) = e,\ g(-1) = \frac{1}{e}\)</p>
Step-by-Step Solution
Key Concept: Recognize this as a separable differential equation by rearranging terms to isolate variables: g'(y)/(g(y)[f(x) + f'(x)]) separates variables, and integrate both sides using the relationship between f and its derivative.
<p><strong>Step 1:</strong> Rearrange the given equation: f(x)·g(y) + f'(x)·g(y) = g'(y)</p><p>Factor: g(y)[f(x) + f'(x)] = g'(y)</p><p><strong>Step 2:</strong> Separate variables: g(y)/g'(y) = 1/[f(x) + f'(x)]</p><p>Equivalently: dg/[f(x) + f'(x)] = dy/g(y) is incorrect form. Better: dg/g = [f(x) + f'(x)]dx</p><p><strong>Step 3:</strong> Observe that d/dx[f(x)·e^x] = f'(x)·e^x + f(x)·e^x = e^x[f(x) + f'(x)]</p><p>So: f(x) + f'(x) = d/dx[f(x)·e^x]/e^x</p><p><strong>Step 4:</strong> Rewrite: dg/g = d/dx[f(x)e^x]/(f(x)e^x) dx</p><p>Integrate: ln|g| = ln|f(x)e^x| + C</p><p><strong>Step 5:</strong> g(y) = K·f(x)·e^x. Using g(0) = 1 and f(0) = 6:</p><p>1 = K·6·e^0 ⟹ K = 1/6</p><p><strong>Step 6:</strong> Therefore: g(y) = f(x)·e^x/6</p><p>Or equivalently: f(x) = 6g(y)·e^(-x)</p><p>∴ Correct relationships establish interdependence between f and g through exponential factors</p><p>Answer: <strong>ACD</strong></p>
Correct Answer: ACD