Find the particular solution that satisfies the given initial condition
$\(\left\)(y^2+y\(\right\))e^{x}y^{\(\prime\)}=y^3+e^{x}y^3;y\(\left\)(0\(\right\))=1$

• A. $\(\ln\]\left\)|y\(\right\)|-\(\frac{1}{y}\)=-e^{-x}+x$
• B. $\(\ln\]\left\)|y\(\right\)|-\(\frac{1}{y}\)=-e^{-x}+x+1$
• C. $\(\ln\[\left\)|\(\frac{y}{y^2}\]\right\)|=-e^{-x}+x$
• D. $\(\ln\[\left\)|\(\frac{y}{y^2}\]\right\)|=-e^{-x}+x+1$

Answer: (A)

Verified step by step guidance

1. Step 1: Start by rewriting the given differential equation \((y^2 + y)e^x y' = y^3 + e^x y^3\) in a more manageable form. Factor out common terms where possible. Notice that \(y^3\) is a common term on the right-hand side.
2. Step 2: Divide through by \((y^2 + y)e^x\) to isolate \(y'\) on one side. This will give you \(y' = \frac{y^3 + e^x y^3}{(y^2 + y)e^x}\). Simplify the expression by factoring and canceling terms where appropriate.
3. Step 3: Recognize that this is a separable differential equation. Rearrange the terms to separate \(y\) and \(x\), so that all \(y\)-dependent terms are on one side and all \(x\)-dependent terms are on the other. This will involve dividing and simplifying the terms.
4. Step 4: Integrate both sides. For the \(y\)-dependent side, you will need to integrate an expression involving \(\frac{1}{y}\) and \(\frac{1}{y^2}\). For the \(x\)-dependent side, you will integrate \(e^{-x}\) and any constants. Use the initial condition \(y(0) = 1\) to solve for the constant of integration.
5. Step 5: After integrating, simplify the resulting expression to match the form of the solution. Combine logarithmic terms and constants as needed. The final solution will involve \(\ln|y|\), \(\frac{1}{y}\), and terms involving \(e^{-x}\) and \(x\).