1.2.3. Linear Differential Equation

Try these first

1. 1.

   What is the order of the differential equation $\frac{d^2y}{dx^2} + 5y = 0$?

   Hint

   Look for the highest derivative present.

2. 2.

   Identify if the equation $\frac{dy}{dx} + 4y = 7$ is linear.

   Hint

   Check if the dependent variable appears only to the first power.

3. 3.

   What is the general form of a first-order linear differential equation?

   • \\( y + P(x) = Q(x) \\)
   • \\( \\frac{dy}{dx} + P(x)y = Q(x) \\)
   • \\( \\frac{dy}{dx} = P(x) + Q(x) \\)

   Hint

   Recall the structure for first-order linear equations.

4. 4.

   Is the equation $y' + 2y = 0$ linear?

   • True
   • False

   Hint

   Check how the dependent variable and its derivative are expressed.

5. 5.

   Given the equation $y''' + 9y = 0$, identify its order and classify if it is linear or not.

   Hint

   Identify the highest derivative to find the order.

6. 6.

   Classify the equation $\frac{d^2y}{dt^2} + 3 \frac{dy}{dt} + 2y = e^t$. Determine its order and if it's homogeneous.

   Hint

   Check the right-hand side for non-homogeneity.