General solution of the Differential equation dy/dx + ay = e^mx is Option 1: (a+m)y = e^mx + Ce^(-ax): where C is an arbitrary constant Option 2: y = e^mx + Ce^(-ax): where C is an arbitrary constant Option 3: (a+m)y = e^mx + C: where C is an arbitrary constant Option 4: y.e^ax = me^mx + C: where C is an arbitrary constant

Answer: (a+m)y = e^mx + Ce^(-ax): where C is an arbitrary constant. Solution: This is linear with integrating factor e^ax. Then ye^ax= e^((a+m)x)dx=(e^((a+m)x))/(a+m)+c. Multiplying by (a+m)e^(-ax) gives (a+m)y=e^mx+Ce^(-ax).

Calculus

Differential Equations

Medium

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General solution of the Differential equation $\frac{dy}{dx} + ay = e^{mx}$ is

$(a+m)y = e^{mx} + Ce^{-ax}$ : where $C$ is an arbitrary constant

$y = e^{mx} + Ce^{-ax}$ : where $C$ is an arbitrary constant

$(a+m)y = e^{mx} + C$ : where $C$ is an arbitrary constant

$y.e^{ax} = me^{mx} + C$ : where $C$ is an arbitrary constant

✅ Correct Option: 1

This is linear with integrating factor $e^{ax}$ . Then $ye^{ax}=\int e^{(a+m)x}dx=\frac{e^{(a+m)x}}{a+m}+c$ . Multiplying by $(a+m)e^{-ax}$ gives $(a+m)y=e^{mx}+Ce^{-ax}$ .

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General solution of the Differential equation $\frac{dy}{dx} + ay = e^{mx}$ is