General solution of the Differential equation \( \displaystyle\frac{dy}{dx}+ay=e^{mx} \) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( (a+m)y=e^{mx}+Ce^{-ax} \): where \( C \) is an arbitrary constant
This is a linear differential equation with P = a, so the integrating factor is
\( \text{I.F.}=e^{\int a\,dx}=e^{ax} \)
Multiplying by the I.F. and integrating (taking \( a+m\ne0 \)),
\( \displaystyle y\,e^{ax}=\int e^{mx}e^{ax}\,dx \)
\( \displaystyle =\frac{e^{(a+m)x}}{a+m}+C_{1} \)
Multiplying both sides by \( (a+m)e^{-ax} \),
\( (a+m)y=e^{mx}+(a+m)C_{1}e^{-ax} \)
\( (a+m)y=e^{mx}+Ce^{-ax} \)
where \( C=(a+m)C_{1} \) is again an arbitrary constant.
Why the other options are incorrect
Option (2) \( y=e^{mx}+Ce^{-ax} \): where \( C \) is an arbitrary constant — Incorrect. The factor \( a+m \) is missing: integrating \( e^{(a+m)x} \) divides it by \( a+m \).
Option (3) \( (a+m)y=e^{mx}+C \): where \( C \) is an arbitrary constant — Incorrect. When the equation is divided by \( e^{ax} \), the constant must also be divided by it, giving \( Ce^{-ax} \).
Option (4) \( y.e^{ax}=me^{mx}+C \): where \( C \) is an arbitrary constant — Incorrect. The right side should be \( \frac{e^{(a+m)x}}{a+m}+C \), not \( me^{mx}+C \).
NCERT Exemplar:
Differential Equations
