Answer :
Let's analyze each equation to determine its type, linearity, order, and identify the independent and dependent variables.
### 1.
[tex]\[ \frac{d^2 x}{d t^2}+k^2 x=0 \][/tex]
- Type: Ordinary Differential Equation (ODE) - because it involves derivatives with respect to a single variable.
- Linearity: Linear - because the dependent variable [tex]\(x\)[/tex] and its derivatives appear to the first power and are not multiplied by each other.
- Order: 2nd order - because the highest derivative present is [tex]\( \frac{d^2 x}{d t^2} \)[/tex].
- Independent Variable: [tex]\( t \)[/tex]
- Dependent Variable: [tex]\( x \)[/tex]
### 2.
[tex]\[ (x^2+y^2) d x+2 x y d y=0 \][/tex]
- Type: Ordinary Differential Equation (ODE) - it can be written as a differential equation in terms of [tex]\( \frac{d y}{d x} \)[/tex].
- Linearity: Nonlinear - because the terms [tex]\(x^2\)[/tex], [tex]\(y^2\)[/tex], and [tex]\(2xy\)[/tex] make the equation nonlinear.
- Order: 1st order - the equation involves only the first differentials [tex]\( dx \)[/tex] and [tex]\( dy \)[/tex], effectively [tex]\( \frac{d y}{d x} \)[/tex].
- Independent Variable: [tex]\( x \)[/tex]
- Dependent Variable: [tex]\( y \)[/tex]
### 3.
[tex]\[ \left(\frac{d^3 w}{d x^3}\right)^2-2\left(\frac{d w}{d x}\right)^4+y w=0 \][/tex]
- Type: Ordinary Differential Equation (ODE)
- Linearity: Nonlinear - the terms [tex]\( \left(\frac{d^3 w}{d x^3}\right)^2 \)[/tex] and [tex]\( \left(\frac{d w}{d x}\right)^4 \)[/tex] make it nonlinear.
- Order: 3rd order - because the highest derivative present is [tex]\( \frac{d^3 w}{d x^3} \)[/tex].
- Independent Variable: [tex]\( x \)[/tex]
- Dependent Variable: [tex]\( w \)[/tex]
### 4.
[tex]\[ \frac{\partial^2 w}{\partial t^2}=a^2 \frac{\partial^2 w}{\partial x^2} \][/tex]
- Type: Partial Differential Equation (PDE) - because it involves partial derivatives with respect to more than one variable.
- Linearity: Linear - because the dependent variable [tex]\( w \)[/tex] and its derivatives appear to the first power and are not multiplied by each other.
- Order: 2nd order - because the highest derivative present is [tex]\( \frac{\partial^2}{\partial t^2} \)[/tex] and [tex]\( \frac{\partial^2}{\partial x^2} \)[/tex].
- Independent Variables: [tex]\( t \)[/tex] and [tex]\( x \)[/tex]
- Dependent Variable: [tex]\( w \)[/tex]
### 5.
[tex]\[ x\left(y^{\prime \prime}\right)^3+\left(y^{\prime}\right)^4-y=0 \][/tex]
- Type: Ordinary Differential Equation (ODE)
- Linearity: Nonlinear - the terms [tex]\( (y^{\prime \prime})^3 \)[/tex] and [tex]\( (y^{\prime})^4 \)[/tex] make it nonlinear.
- Order: 2nd order - because the highest derivative present is [tex]\( y^{\prime \prime} \)[/tex].
- Independent Variable: [tex]\( x \)[/tex]
- Dependent Variable: [tex]\( y \)[/tex]
By this detailed analysis, we categorize each equation with the corresponding characteristics, crucial for solving or understanding the nature of the differential equations.
### 1.
[tex]\[ \frac{d^2 x}{d t^2}+k^2 x=0 \][/tex]
- Type: Ordinary Differential Equation (ODE) - because it involves derivatives with respect to a single variable.
- Linearity: Linear - because the dependent variable [tex]\(x\)[/tex] and its derivatives appear to the first power and are not multiplied by each other.
- Order: 2nd order - because the highest derivative present is [tex]\( \frac{d^2 x}{d t^2} \)[/tex].
- Independent Variable: [tex]\( t \)[/tex]
- Dependent Variable: [tex]\( x \)[/tex]
### 2.
[tex]\[ (x^2+y^2) d x+2 x y d y=0 \][/tex]
- Type: Ordinary Differential Equation (ODE) - it can be written as a differential equation in terms of [tex]\( \frac{d y}{d x} \)[/tex].
- Linearity: Nonlinear - because the terms [tex]\(x^2\)[/tex], [tex]\(y^2\)[/tex], and [tex]\(2xy\)[/tex] make the equation nonlinear.
- Order: 1st order - the equation involves only the first differentials [tex]\( dx \)[/tex] and [tex]\( dy \)[/tex], effectively [tex]\( \frac{d y}{d x} \)[/tex].
- Independent Variable: [tex]\( x \)[/tex]
- Dependent Variable: [tex]\( y \)[/tex]
### 3.
[tex]\[ \left(\frac{d^3 w}{d x^3}\right)^2-2\left(\frac{d w}{d x}\right)^4+y w=0 \][/tex]
- Type: Ordinary Differential Equation (ODE)
- Linearity: Nonlinear - the terms [tex]\( \left(\frac{d^3 w}{d x^3}\right)^2 \)[/tex] and [tex]\( \left(\frac{d w}{d x}\right)^4 \)[/tex] make it nonlinear.
- Order: 3rd order - because the highest derivative present is [tex]\( \frac{d^3 w}{d x^3} \)[/tex].
- Independent Variable: [tex]\( x \)[/tex]
- Dependent Variable: [tex]\( w \)[/tex]
### 4.
[tex]\[ \frac{\partial^2 w}{\partial t^2}=a^2 \frac{\partial^2 w}{\partial x^2} \][/tex]
- Type: Partial Differential Equation (PDE) - because it involves partial derivatives with respect to more than one variable.
- Linearity: Linear - because the dependent variable [tex]\( w \)[/tex] and its derivatives appear to the first power and are not multiplied by each other.
- Order: 2nd order - because the highest derivative present is [tex]\( \frac{\partial^2}{\partial t^2} \)[/tex] and [tex]\( \frac{\partial^2}{\partial x^2} \)[/tex].
- Independent Variables: [tex]\( t \)[/tex] and [tex]\( x \)[/tex]
- Dependent Variable: [tex]\( w \)[/tex]
### 5.
[tex]\[ x\left(y^{\prime \prime}\right)^3+\left(y^{\prime}\right)^4-y=0 \][/tex]
- Type: Ordinary Differential Equation (ODE)
- Linearity: Nonlinear - the terms [tex]\( (y^{\prime \prime})^3 \)[/tex] and [tex]\( (y^{\prime})^4 \)[/tex] make it nonlinear.
- Order: 2nd order - because the highest derivative present is [tex]\( y^{\prime \prime} \)[/tex].
- Independent Variable: [tex]\( x \)[/tex]
- Dependent Variable: [tex]\( y \)[/tex]
By this detailed analysis, we categorize each equation with the corresponding characteristics, crucial for solving or understanding the nature of the differential equations.
