Differential Equations

Step 1: Identifying the order of the differential equation.
The given equation is: The highest derivative of with respect to in this equation is , which is the second derivative.

Step 2: Conclusion.
Since the highest derivative is the second derivative, the order of the differential equation is 2. Hence, the statement is true.

Step 1: Write the equation in standard form.
The given equation is: We divide the entire equation by to simplify it: This is a linear first-order differential equation of the form: where and .

Step 2: Find the integrating factor.
The integrating factor is given by:

Step 3: Multiply through by the integrating factor.
Multiplying the entire equation by , we get:

Step 4: Simplify the equation.
The left-hand side is the derivative of , so we have:

Step 5: Integrate both sides.
Integrating both sides with respect to , we get: where is the constant of integration.

Step 6: Solve for .
Thus, the general solution is:

Step 7: Conclusion.
The general solution to the differential equation is:

Step 1: Separate variables.
We can separate the variables and to integrate both sides:

Step 2: Integrate both sides.
The integral of is , and the integral of is , so we get: where is the constant of integration.

Step 3: Solve for .
Thus, the general solution is:

Step 4: Conclusion.
The general solution to the differential equation is:

Step 1: Identifying the integrating factor.
The differential equation is: This is a first-order linear differential equation of the form , where and . The integrating factor is given by: Since , we have:

Step 2: Conclusion.
Thus, the integrating factor is , which makes the statement true.

Step 1: Understanding arbitrary constants.
For a differential equation of order , the number of arbitrary constants in its particular solution is .

Step 2: Conclusion.
Thus, for a third-order differential equation, the number of arbitrary constants is 3.