Step-by-step Solution:

This problem requires solving a first-order ordinary differential equation with a given initial value.

1. Identify the type of Differential Equation (DE):
The equation is \((y^{2}+3xy)dx+(x^{2}+xy)dy=0\). This is in the form \(M(x,y)dx + N(x,y)dy=0\).
We check if it's an exact equation by testing if \(\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}\).
\[M = y^2 + 3xy \implies \frac{\partial M}{\partial y} = 2y + 3x\]\[N = x^2 + xy \implies \frac{\partial N}{\partial x} = 2x + y\]Since \(2y+3x \neq 2x+y\), the equation is not exact. We can solve it by finding an integrating factor.

2. Find the Integrating Factor (I.F.):
We compute the term \(\frac{1}{N}(\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x})\):
\[\frac{1}{N}(\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x}) = \frac{(2y+3x)-(2x+y)}{x^2+xy} = \frac{y+x}{x(x+y)} = \frac{1}{x}\]Since this result is a function of \(x\) alone, the integrating factor \(\mu(x)\) is:
\[\mu(x) = e^{\int \frac{1}{x} dx} = e^{\ln x} = x\]
3. Make the Equation Exact and Solve:
Multiply the original DE by the I.F. \((x)\):
\[x(y^2+3xy)dx + x(x^2+xy)dy=0\]\[(xy^2+3x^2y)dx + (x^3+x^2y)dy=0\]This new equation is exact. Its solution \(F(x,y)=C\) is found by integrating the \(dx\) term with respect to \(x\):
\[F(x,y) = \int (xy^2+3x^2y)dx = \frac{x^2y^2}{2} + x^3y + g(y)\]To find \(g(y)\), we differentiate this result with respect to \(y\) and set it equal to the new \(N\) term:
\[\frac{\partial F}{\partial y} = x^2y + x^3 + g'(y) = x^3+x^2y\]This implies \(g'(y)=0\), so \(g(y)\) is a constant. The general solution is:
\[\frac{x^2y^2}{2} + x^3y = C\]
4. Apply the Initial Condition:
We are given the initial condition \(y=1\) when \(x=1\). We substitute these values to find the constant \(C\).
\[\frac{(1)^2(1)^2}{2} + (1)^3(1) = C \implies \frac{1}{2} + 1 = C \implies C = \frac{3}{2}\]The particular solution is therefore:
\[\frac{x^2y^2}{2} + x^3y = \frac{3}{2} \quad \text{or} \quad x^2y^2 + 2x^3y - 3 = 0\]
5. Conclusion and Analysis of Options:
The correct solution to the problem as stated is the implicit equation \(x^2y^2 + 2x^3y = 3\). This does not simplify to any of the provided options.

Furthermore, we can check if any of the options satisfy the initial condition \(y(1)=1\):
- Option A: \(y(1) = 1^2/(1+1) = 1/2 \neq 1\)
- Option B: \(y(1) = (1+1)/1^2 = 2 \neq 1\)
- Option C: \(y(1) = 1+1 = 2 \neq 1\)
- Option D: \(y(1) = 1/(1+3) = 1/4 \neq 1\)
None of the options are consistent with the problem's initial condition. This indicates an error in the problem statement or the options.

Result:
While the provided solution key indicates option D, it is inconsistent with the problem statement. The correct implicit solution is \(x^2y^2 + 2x^3y = 3\).