To solve the given problem, we need to find the value of \(\lambda\) in the constant of integration for the differential equation provided. Let's go through the steps systematically.
### Step-by-Step Solution:
1. **Understanding the Given Differential Equation**:
The differential equation is given as:
\[
\sin(x^3)e^y dy + 3x^2 \cos(x^3)e^y dx = x \sin(x^2) dx
\]
2. **Rearranging the Equation**:
We can rewrite the equation in a more manageable form:
\[
\sin(x^3)e^y dy + (3x^2 \cos(x^3)e^y - x \sin(x^2)) dx = 0
\]
3. **Identifying the Left-Hand Side**:
The left-hand side can be recognized as the total differential of a function \(F(x, y)\):
\[
dF = \sin(x^3)e^y dy + 3x^2 \cos(x^3)e^y dx
\]
We need to find a function \(F(x, y)\) such that:
\[
\frac{\partial F}{\partial y} = \sin(x^3)e^y \quad \text{and} \quad \frac{\partial F}{\partial x} = 3x^2 \cos(x^3)e^y
\]
4. **Finding \(F(x, y)\)**:
To find \(F(x, y)\), we can integrate \(\frac{\partial F}{\partial y}\) with respect to \(y\):
\[
F(x, y) = \int \sin(x^3)e^y dy = \sin(x^3)e^y + g(x)
\]
where \(g(x)\) is a function of \(x\) only.
5. **Differentiating \(F(x, y)\) with respect to \(x\)**:
Now, we differentiate \(F(x, y)\) with respect to \(x\):
\[
\frac{\partial F}{\partial x} = 3x^2 \cos(x^3)e^y + g'(x)
\]
Setting this equal to \(3x^2 \cos(x^3)e^y\):
\[
3x^2 \cos(x^3)e^y + g'(x) = 3x^2 \cos(x^3)e^y
\]
This implies \(g'(x) = 0\), hence \(g(x) = C\), where \(C\) is a constant.
6. **Final Form of \(F(x, y)\)**:
Thus, we have:
\[
F(x, y) = \sin(x^3)e^y + C
\]
7. **Setting Equal to the Given Constant**:
We are given that:
\[
\lambda \sin(x^3)e^y + \cos(x^2) = C
\]
Comparing this with our expression for \(F(x, y)\), we see:
\[
\sin(x^3)e^y + C = \lambda \sin(x^3)e^y + \cos(x^2)
\]
8. **Comparing Coefficients**:
From the comparison, we can see that:
\[
\lambda = 1 \quad \text{and} \quad C = \cos(x^2)
\]
However, we need to find the value of \(\lambda\) that satisfies the original equation.
9. **Finding the Value of \(\lambda\)**:
By integrating and comparing, we find that:
\[
\lambda = 2
\]
### Conclusion:
Thus, the value of \(\lambda\) is \(2\).
