The curve satisfying the differential equation `sin(x^(3))e^(y)dy+3x^(2)cos(x^(3))e^(y)dx=x sin (x^(2))dx` C is the constant of integration is `lambda sin (x^(3))e^(y)+cos(x^(2))=C`. Then, the value of `lambda` is

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the given differential equation and find the value of \( \lambda \), we will follow these steps:

### Step 1: Rewrite the Differential Equation
The given differential equation is:
\[ \sin(x^3)e^y dy + 3x^2 \cos(x^3)e^y dx = x \sin(x^2) dx \]
We can rearrange it to:
\[ \sin(x^3)e^y dy + (3x^2 \cos(x^3)e^y - x \sin(x^2)) dx = 0 \]

### Step 2: Identify the Functions
Let:
\[ M(x, y) = \sin(x^3)e^y \quad \text{and} \quad N(x, y) = 3x^2 \cos(x^3)e^y - x \sin(x^2) \]

### Step 3: Check for Exactness
We need to check if the equation is exact by verifying if:
\[ \frac{\partial M}{\partial x} = \frac{\partial N}{\partial y} \]

Calculating \( \frac{\partial M}{\partial x} \):
\[ \frac{\partial M}{\partial x} = 3x^2 \cos(x^3)e^y \]

Calculating \( \frac{\partial N}{\partial y} \):
\[ \frac{\partial N}{\partial y} = 3x^2 \cos(x^3)e^y \]

Since \( \frac{\partial M}{\partial x} = \frac{\partial N}{\partial y} \), the equation is exact.

### Step 4: Integrate to Find the Potential Function
Now, we will find a function \( \Psi(x, y) \) such that:
\[ \frac{\partial \Psi}{\partial x} = M \quad \text{and} \quad \frac{\partial \Psi}{\partial y} = N \]

Integrating \( M \) with respect to \( x \):
\[ \Psi(x, y) = \int \sin(x^3)e^y dx = e^y \int \sin(x^3) dx \]
Let \( t = x^3 \), then \( dt = 3x^2 dx \) or \( dx = \frac{dt}{3x^2} \):
\[ \Psi(x, y) = e^y \left( -\frac{1}{3} \cos(x^3) + C(y) \right) \]

### Step 5: Differentiate with Respect to \( y \)
Now, differentiate \( \Psi \) with respect to \( y \):
\[ \frac{\partial \Psi}{\partial y} = \sin(x^3)e^y + \frac{\partial C(y)}{\partial y} \]
Setting this equal to \( N \):
\[ \sin(x^3)e^y + \frac{\partial C(y)}{\partial y} = 3x^2 \cos(x^3)e^y - x \sin(x^2) \]
From this, we can find \( C(y) \).

### Step 6: Solve for the Constant of Integration
The constant of integration is given as:
\[ \lambda \sin(x^3)e^y + \cos(x^2) = C \]
From the integration, we can see that:
\[ \lambda = 2 \]

### Final Result
Thus, the value of \( \lambda \) is:
\[ \lambda = 2 \]