Question

The solution of the differential equation $\frac{dx}{dy}+Px=Q$, where P and Q are constants or functions of y, is given by ________.

• A. $xe^{\int Pdx}=\int Qe^{\int Pdx}dx+c$
• B. $ye^{\int Pdy}=\int Qe^{\int Pdy}dy+c$
• C. $ye^{\int Pdx}=\int Qe^{\int Pdx}dx+c$
• D. $xe^{\int Pdy}=\int Qe^{\int Pdy}dy+c$

Hint:

If the equation is $\frac{dx}{dy}$, solve for $x$ as a function of $y$.

Answer 1

The Correct Option is D

Step 1: Concept
This is a linear differential equation of the form $\frac{dx}{dy} + Px = Q$.
Step 2: Analysis
For this form, the integrating factor ($IF$) is calculated with respect to $y$ because $P$ is a function of $y$. $IF = e^{\int P dy}$.
Step 3: Calculation
The general solution is given by: $x \cdot (IF) = \int (Q \cdot IF) dy + c$. Substituting $IF$: $x e^{\int P dy} = \int Q e^{\int P dy} dy + c$.
Step 4: Conclusion
Hence, the correct solution format is option (D).
Final Answer:(D)