Comparing Separable and Integrating Factor Methods

00:00 8 marks

Consider the following differential equations:

(a) \(\frac{dy}{dx} = xy\).

(b) \(\frac{dy}{dx} – y = e^x\).

For part (a), determine if the differential equation is separable and solve it if so. For part (b), solve the differential equation using the integrating factor method.

Part a

5 minutes 4 marks

Question

Determine whether the differential equation \(\frac{dy}{dx} = xy\) is separable. If so, separate the variables and solve for \(y\) in terms of \(x\).

Hint Hint

Hint

Rewrite the equation to isolate the \(y\) terms and \(dy\) on one side and the \(x\) terms and \(dx\) on the other. Then integrate both sides.

Answer Answer
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Complete Answer

\(y = Ce^{\frac{x^2}{2}}\)

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Part b

5 minutes 4 marks

Question

Solve the differential equation \(\frac{dy}{dx} – y = e^x\) using the integrating factor method.

Hint Hint

Hint

Rewrite the equation in the standard linear form \(\frac{dy}{dx} + P(x)y = Q(x)\). Identify \(P(x)\) and calculate the integrating factor \(\mu(x) = e^{\int P(x)\,dx}\).

Answer Answer
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Complete Answer

\(y = e^x (x + C)\)

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