Find the general solution of the differential equation: $x^2{y}^{\prime }+x(x+2)y=e^x$.

• A. $y=\frac{e^x}{x}+C$
• B. $y=x^2{e}^x+C$
• C. $y=\frac{1}{x^2}[\int x{e}^{x}dx+C]$
• D. $y=e^x+C{x}^2$

Answer: (C)

Verified step by step guidance

1. Step 1: Recognize that the given differential equation is a first-order linear differential equation of the form x²y' + x(x + 2)y = e^x. Rewrite it in standard form by dividing through by x² to isolate y': y' + (x + 2)/x y = e^x/x².
2. Step 2: Identify the integrating factor for the equation. The integrating factor is derived from the coefficient of y in the standard form, which is (x + 2)/x. Compute the integrating factor as e^(∫(x + 2)/x dx). Break this integral into simpler parts: ∫(x + 2)/x dx = ∫1 dx + ∫2/x dx = x + 2ln|x|.
3. Step 3: Write the integrating factor as e^(x + 2ln|x|). Simplify this expression using properties of exponents: e^(x + 2ln|x|) = e^x * e^(ln|x²|) = e^x * x².
4. Step 4: Multiply through the differential equation by the integrating factor (e^x * x²) to make the left-hand side an exact derivative. The equation becomes: d/dx[(e^x * x² * y)] = e^x * x.
5. Step 5: Integrate both sides with respect to x. The left-hand side integrates to e^x * x² * y, and the right-hand side requires integration of e^x * x. After integrating, solve for y to find the general solution: y = (1/x²) [∫x e^x dx + C], where C is the constant of integration.