Directions: In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as
Solution of the differential equation
`(dy)/(dx)=e^(3x-2y)+x^2e^(-2y)` is `(e^(2y))/(2)=(e^(3x))/(3)+(x^2)/(2)+C`
Reason (R) :
`(dy)/(dx)=e^(3x-2y)+x^2e^(-2y)`
`(dy)/(dx)=e^(-2y)(e^(3x)+x^2)`
separating the variables
`e^(2y)dy=(e^(3x)+x^2)dx`
`int e^(2y)dy=int(e^(3x)+x^2)dx`
`(e^(2y))/(2)=(e^(3x))/(3)+(x^3)/(3)+C`.

Options

A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true but R is NOT the correct explanation of A
C. A is true but R is false
D. A is false but R is True

Answer

To solve the differential equation given in the question, we will follow the steps of separation of variables and integration. Let's go through the solution step by step.

### Step 1: Write the differential equation
The given differential equation is:
\[ \frac{dy}{dx} = e^{3x - 2y} + x^2 e^{-2y} \]

### Step 2: Factor out \( e^{-2y} \)
We can rewrite the right-hand side by factoring out \( e^{-2y} \):
\[ \frac{dy}{dx} = e^{-2y} \left( e^{3x} + x^2 \right) \]

### Step 3: Separate the variables
Now, we can separate the variables \( y \) and \( x \):
\[ e^{2y} dy = (e^{3x} + x^2) dx \]

### Step 4: Integrate both sides
Next, we will integrate both sides:
\[ \int e^{2y} dy = \int (e^{3x} + x^2) dx \]

### Step 5: Perform the integration
The left-hand side integrates to:
\[ \frac{e^{2y}}{2} \]
The right-hand side integrates to:
\[ \int e^{3x} dx = \frac{e^{3x}}{3} + \int x^2 dx = \frac{e^{3x}}{3} + \frac{x^3}{3} \]

### Step 6: Combine the results
Combining both results gives us:
\[ \frac{e^{2y}}{2} = \frac{e^{3x}}{3} + \frac{x^3}{3} + C \]

### Step 7: Rearranging the equation
We can rearrange the equation to express it in a more standard form:
\[ e^{2y} = \frac{2e^{3x}}{3} + \frac{2x^3}{3} + 2C \]

### Conclusion
The final solution of the differential equation is:
\[ \frac{e^{2y}}{2} = \frac{e^{3x}}{3} + \frac{x^3}{3} + C \]