Worksheet #1: Complete the table with details for 10 differential equations.

Let's analyze each differential equation in the worksheet and fill out the table with the correct information:

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Definitions:

- Order: The highest derivative present.
- Degree: The power of the highest derivative when the equation is a polynomial in derivatives (after eliminating radicals and fractions).
- Linearity: A DE is linear if the dependent variable and its derivatives appear to the first power and are not multiplied together. Otherwise, it’s nonlinear.
- Independent Variables: The variables with respect to which differentiation occurs.
- Dependent Variables: The variables being differentiated (e.g., y, s, v, z, etc.).
- Ordinary or Partial: Ordinary if only one independent variable; partial if more than one.

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Now, let’s go through each problem:

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1.

$$
\left[\frac{d^2y}{dx^2}\right]^3 = k\left[1 + \left(\frac{dy}{dx}\right)^4\right]
$$

- Order: 2 (highest derivative is $ \frac{d^2y}{dx^2} $)
- Degree: 3 (since $ \left(\frac{d^2y}{dx^2}\right)^3 $ is raised to the 3rd power)
- Linearity: Nonlinear (due to powers higher than 1 on derivatives)
- Independent Variable: $ x $
- Dependent Variable: $ y $
- Type: Ordinary (only one independent variable)

✔ Answer:
| Order | Degree | Linearity | Independent Var | Dependent Var | Type |
|-------|--------|-----------|------------------|----------------|--------|
| 2 | 3 | Nonlinear | x | y | Ordinary |

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2.

$$
\frac{d^3y}{dx^3} = \sqrt{\frac{dy}{dx}}
$$

- Order: 3
- Degree: 2 (square root → raise both sides: $ \left(\frac{d^3y}{dx^3}\right)^2 = \frac{dy}{dx} $, so degree = 2)
- Linearity: Nonlinear (because of square root)
- Independent Variable: $ x $
- Dependent Variable: $ y $
- Type: Ordinary

✔ Answer:
| Order | Degree | Linearity | Independent Var | Dependent Var | Type |
|-------|--------|-----------|------------------|----------------|--------|
| 3 | 2 | Nonlinear | x | y | Ordinary |

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3.

$$
\sqrt{\frac{d^2y}{dx^2}} = 3\frac{dy}{dx}
$$

- Order: 2
- Degree: 2 (square both sides: $ \frac{d^2y}{dx^2} = 9\left(\frac{dy}{dx}\right)^2 $, so degree = 2)
- Linearity: Nonlinear (due to square root and squared derivative)
- Independent Variable: $ x $
- Dependent Variable: $ y $
- Type: Ordinary

✔ Answer:
| Order | Degree | Linearity | Independent Var | Dependent Var | Type |
|-------|--------|-----------|------------------|----------------|--------|
| 2 | 2 | Nonlinear | x | y | Ordinary |

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4.

$$
\left(\frac{d^3s}{dt^3}\right)^2 + \left(\frac{d^2s}{dt^2}\right)^3 = s - 3t
$$

- Order: 3
- Degree: 2 (highest derivative is $ \frac{d^3s}{dt^3} $, raised to power 2)
- Linearity: Nonlinear (powers >1 on derivatives)
- Independent Variable: $ t $
- Dependent Variable: $ s $
- Type: Ordinary

✔ Answer:
| Order | Degree | Linearity | Independent Var | Dependent Var | Type |
|-------|--------|-----------|------------------|----------------|--------|
| 3 | 2 | Nonlinear | t | s | Ordinary |

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5.

$$
\frac{dr}{d\theta} = \sqrt{r\theta}
$$

- Order: 1
- Degree: 2 (square both sides: $ \left(\frac{dr}{d\theta}\right)^2 = r\theta $)
- Linearity: Nonlinear (because $ r $ appears under square root and multiplied by $ \theta $)
- Independent Variable: $ \theta $
- Dependent Variable: $ r $
- Type: Ordinary

✔ Answer:
| Order | Degree | Linearity | Independent Var | Dependent Var | Type |
|-------|--------|-----------|------------------|----------------|--------|
| 1 | 2 | Nonlinear | $ \theta $ | r | Ordinary |

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6.

$$
\frac{d^2y}{dx^2} = \sqrt{1 + \left(\frac{dy}{dx}\right)^4}
$$

- Order: 2
- Degree: 2 (square both sides: $ \left(\frac{d^2y}{dx^2}\right)^2 = 1 + \left(\frac{dy}{dx}\right)^4 $, so degree = 2)
- Linearity: Nonlinear (due to square root and high power)
- Independent Variable: $ x $
- Dependent Variable: $ y $
- Type: Ordinary

✔ Answer:
| Order | Degree | Linearity | Independent Var | Dependent Var | Type |
|-------|--------|-----------|------------------|----------------|--------|
| 2 | 2 | Nonlinear | x | y | Ordinary |

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7.

$$
\frac{\partial^2 v}{\partial y^2} = \frac{\partial v}{\partial x} + 1
$$

- Order: 2 (highest derivative is second order)
- Degree: 1 (no powers or radicals)
- Linearity: Linear (all terms are linear in $ v $ and its derivatives)
- Independent Variables: $ x, y $
- Dependent Variable: $ v $
- Type: Partial (two independent variables)

✔ Answer:
| Order | Degree | Linearity | Independent Var | Dependent Var | Type |
|-------|--------|-----------|------------------|----------------|--------|
| 2 | 1 | Linear | x, y | v | Partial |

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8.

$$
\frac{\partial^4 z}{\partial y^4} + \left(\frac{\partial^2 z}{\partial x \partial y}\right)^6 = x
$$

- Order: 4 (highest derivative is fourth order)
- Degree: 6 (the mixed derivative is raised to the 6th power)
- Linearity: Nonlinear (due to $ (\cdot)^6 $)
- Independent Variables: $ x, y $
- Dependent Variable: $ z $
- Type: Partial

✔ Answer:
| Order | Degree | Linearity | Independent Var | Dependent Var | Type |
|-------|--------|-----------|------------------|----------------|--------|
| 4 | 6 | Nonlinear | x, y | z | Partial |

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9.

$$
x^2 \frac{d^2y}{dx^2} + 2x \frac{dy}{dx} + y = x^2 + 2
$$

- Order: 2
- Degree: 1 (all derivatives are to the first power)
- Linearity: Linear (dependent variable and derivatives appear linearly)
- Independent Variable: $ x $
- Dependent Variable: $ y $
- Type: Ordinary

✔ Answer:
| Order | Degree | Linearity | Independent Var | Dependent Var | Type |
|-------|--------|-----------|------------------|----------------|--------|
| 2 | 1 | Linear | x | y | Ordinary |

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10.

$$
\left(\frac{d^3s}{dt^3}\right)^2 + s \frac{ds}{dt} + 2st = 0
$$

- Order: 3
- Degree: 2 (highest derivative $ \frac{d^3s}{dt^3} $ is squared)
- Linearity: Nonlinear (due to $ \left(\frac{d^3s}{dt^3}\right)^2 $ and $ s \frac{ds}{dt} $ — product of dependent variable and derivative)
- Independent Variable: $ t $
- Dependent Variable: $ s $
- Type: Ordinary

✔ Answer:
| Order | Degree | Linearity | Independent Var | Dependent Var | Type |
|-------|--------|-----------|------------------|----------------|--------|
| 3 | 2 | Nonlinear | t | s | Ordinary |

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✔ Final Table:

| # | Differential Equation | Order | Degree | Linearity | Independent Variables | Dependent Variables | Ordinary or Partial |
|---|------------------------|-------|--------|-----------|------------------------|----------------------|---------------------|
| 1 | $\left[\frac{d^2y}{dx^2}\right]^3 = k\left[1 + \left(\frac{dy}{dx}\right)^4\right]$ | 2 | 3 | Nonlinear | $x$ | $y$ | Ordinary |
| 2 | $\frac{d^3y}{dx^3} = \sqrt{\frac{dy}{dx}}$ | 3 | 2 | Nonlinear | $x$ | $y$ | Ordinary |
| 3 | $\sqrt{\frac{d^2y}{dx^2}} = 3\frac{dy}{dx}$ | 2 | 2 | Nonlinear | $x$ | $y$ | Ordinary |
| 4 | $\left(\frac{d^3s}{dt^3}\right)^2 + \left(\frac{d^2s}{dt^2}\right)^3 = s - 3t$ | 3 | 2 | Nonlinear | $t$ | $s$ | Ordinary |
| 5 | $\frac{dr}{d\theta} = \sqrt{r\theta}$ | 1 | 2 | Nonlinear | $\theta$ | $r$ | Ordinary |
| 6 | $\frac{d^2y}{dx^2} = \sqrt{1 + \left(\frac{dy}{dx}\right)^4}$ | 2 | 2 | Nonlinear | $x$ | $y$ | Ordinary |
| 7 | $\frac{\partial^2 v}{\partial y^2} = \frac{\partial v}{\partial x} + 1$ | 2 | 1 | Linear | $x, y$ | $v$ | Partial |
| 8 | $\frac{\partial^4 z}{\partial y^4} + \left(\frac{\partial^2 z}{\partial x \partial y}\right)^6 = x$ | 4 | 6 | Nonlinear | $x, y$ | $z$ | Partial |
| 9 | $x^2 \frac{d^2y}{dx^2} + 2x \frac{dy}{dx} + y = x^2 + 2$ | 2 | 1 | Linear | $x$ | $y$ | Ordinary |
| 10 | $\left(\frac{d^3s}{dt^3}\right)^2 + s \frac{ds}{dt} + 2st = 0$ | 3 | 2 | Nonlinear | $t$ | $s$ | Ordinary |

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