4.1 EXERCISES Determine the order of the following differential equations. 1. \( y^{\prime}+y=3 y^{2} \) 2. \( \left(y^{\prime}\right)^{2}=y^{\prime}+2 y \) 3. \( y^{\prime \prime}+y^{\prime \prime} y^{\prime}=3 x^{2} \) 4. \( y^{\prime}=y^{\prime \prime}+3 t^{2} \) 5. \( \frac{d y}{d t}=t \) 6. \( \frac{d y}{d x}+\frac{d^{2} y}{d x^{2}}=3 x^{4} \) 7. \( \left(\frac{d y}{d t}\right)^{2}+8 \frac{d y}{d t}+3 y=4 t \)

1. In the equation \( y^{\prime} + y = 3y^2 \), the highest derivative is \( y^{\prime} \) (first order). **Order: 1** 2. In the equation \( \left(y^{\prime}\right)^2 = y^{\prime} + 2y \), although \( y^{\prime} \) is squared, it is still a first derivative. **Order: 1** 3. In the equation \( y^{\prime \prime} + y^{\prime \prime}y^{\prime} = 3x^2 \), the highest derivative that appears is \( y^{\prime \prime} \) (second order). **Order: 2** 4. In the equation \( y^{\prime} = y^{\prime \prime} + 3t^2 \), the highest derivative is \( y^{\prime \prime} \) (second order). **Order: 2** 5. In the equation \( \frac{dy}{dt} = t \), the only derivative present is \( \frac{dy}{dt} \) (first order). **Order: 1** 6. In the equation \( \frac{dy}{dx} + \frac{d^2y}{dx^2} = 3x^4 \), the highest derivative is \( \frac{d^2y}{dx^2} \) (second order). **Order: 2** 7. In the equation \( \left(\frac{dy}{dt}\right)^2 + 8\frac{dy}{dt} + 3y = 4t \), the highest derivative is \( \frac{dy}{dt} \) (first order). **Order: 1**