Question

The general solution of the differential equation of the type \(\frac{dy}{dx} + py = Q\) is:
(where \(P\) and \(Q\) are functions of \(x\) only or constant)

• A. \(y\,e^{\int p\,dy} = \int \left(Q\,e^{\int p\,dy}\right) dy + c\) : (where \(c\) is an arbitrary constant)
• B. \(y\,e^{\int p\,dx} = \int \left(Q\,e^{\int p\,dx}\right) dx + c\) : (where \(c\) is an arbitrary constant)
• C. \(x\,e^{\int p\,dy} = \int \left(Q\,e^{\int p\,dy}\right) dy + c\) : (where \(c\) is an arbitrary constant)
• D. \(x\,e^{\int p\,dx} = \int \left(Q\,e^{\int p\,dx}\right) dx + c\) (where \(c\) is an arbitrary constant)

Hint:

Multiply by the integrating factor \(e^{\int P\,dx}\) and integrate with respect to \(x\).

Answer 1

The Correct Option is B

Step 1: Recognise the form.
The equation \(\frac{dy}{dx} + Py = Q\) is a first order linear equation. Here \(P\) and \(Q\) depend on \(x\) only, so \(x\) is the independent variable and \(y\) is the dependent one.

Step 2: Find the integrating factor.
The integrating factor is \(\text{I.F.} = e^{\int P\,dx}\). Multiplying the equation by it makes the left side a perfect derivative: \[ \frac{d}{dx}\left(y\,e^{\int P\,dx}\right) = Q\,e^{\int P\,dx} \]

Step 3: Integrate.
Integrating both sides with respect to \(x\): \[ y\,e^{\int P\,dx} = \int Q\,e^{\int P\,dx}\,dx + c \]

Step 4: Check the options.
Options 1 and 3 integrate with respect to \(y\), which is wrong since \(P\) and \(Q\) depend on \(x\). Options 3 and 4 have \(x\) on the left, which is the form for \(\frac{dx}{dy} + Px = Q\) with \(P, Q\) functions of \(y\). Option 2 matches our result.

Final Answer:
The general solution is option 2.