ไหนสดงวิธีทำหรืออธีบายอย่างละเอียด 1. จงหาผลเฉลยหั่ไไปของสมการต่อไปนี้ 1) \( \frac{d y}{d x}=\frac{3 x^{2}+4 x}{2 y-2} \) 2) \( \frac{d y}{d x}=\frac{x y^{2}-4 y^{2}}{x^{2} y^{2}-3 x^{3}} \) 3) \( \frac{d y}{d x}=\frac{x y^{2}+3 x}{x^{2} y+y} \)

Alright, I need to explain in detail how to solve the given differential equations. Let's take them one by one. **1) \( \frac{d y}{d x} = \frac{3x^{2} + 4x}{2y - 2} \)** First, I notice that this is a first-order ordinary differential equation (ODE). The equation is separable because the variables \( x \) and \( y \) can be separated on either side of the equation. To solve it, I'll rearrange the equation to separate the variables. Starting with: \[ \frac{d y}{d x} = \frac{3x^{2} + 4x}{2y - 2} \] I can rewrite this as: \[ (2y - 2) \, dy = (3x^{2} + 4x) \, dx \] Now, I'll integrate both sides: \[ \int (2y - 2) \, dy = \int (3x^{2} + 4x) \, dx \] Calculating the integrals: \[ \int 2y \, dy - \int 2 \, dy = \int 3x^{2} \, dx + \int 4x \, dx \] \[ y^{2} - 2y = x^{3} + 2x^{2} + C \] Where \( C \) is the constant of integration. This is the general solution to the differential equation. **2) \( \frac{d y}{d x} = \frac{x y^{2} - 4 y^{2}}{x^{2} y^{2} - 3 x^{3}} \)** This equation looks a bit more complex. Let's see if it's separable. The numerator and denominator both have terms involving \( y^{2} \) and \( x \). I can factor out \( y^{2} \) from the numerator and \( x^{2} \) from the denominator: \[ \frac{d y}{d x} = \frac{y^{2}(x - 4)}{x^{2}(y^{2} - 3x)} \] This doesn't seem to separate easily. Maybe I can use substitution. Let's try substituting \( v = y^{2} \), which implies \( \frac{d v}{d x} = 2y \frac{d y}{d x} \). Substituting into the equation: \[ \frac{1}{2} \frac{d v}{d x} = \frac{v(x - 4)}{x^{2}(v - 3x)} \] This still looks complicated. Perhaps another substitution or method like integrating factors could work, but it's getting messy. Maybe I should consider if the equation is exact or if it can be transformed into a linear equation. Alternatively, I might need to use numerical methods or look for an integrating factor that simplifies the equation. **3) \( \frac{d y}{d x} = \frac{x y^{2} + 3x}{x^{2} y + y} \)** Let's analyze this equation. The numerator and denominator both have terms involving \( x \) and \( y \). I can factor out \( x \) from the numerator and \( y \) from the denominator: \[ \frac{d y}{d x} = \frac{x(y^{2} + 3)}{y(x^{2} + 1)} \] This simplifies to: \[ \frac{d y}{d x} = \frac{x(y^{2} + 3)}{y(x^{2} + 1)} \] To separate the variables, I can rearrange the equation: \[ \frac{y}{y^{2} + 3} \, dy = \frac{x}{x^{2} + 1} \, dx \] Now, I'll integrate both sides: \[ \int \frac{y}{y^{2} + 3} \, dy = \int \frac{x}{x^{2} + 1} \, dx \] Calculating the integrals: \[ \frac{1}{2} \ln(y^{2} + 3) = \frac{1}{2} \ln(x^{2} + 1) + C \] Multiplying both sides by 2: \[ \ln(y^{2} + 3) = \ln(x^{2} + 1) + C \] Exponentiating both sides to eliminate the natural logarithm: \[ y^{2} + 3 = e^{C}(x^{2} + 1) \] Let \( e^{C} = K \), where \( K \) is a constant: \[ y^{2} = K(x^{2} + 1) - 3 \] This is the general solution to the differential equation.