## 1. A differential equation is an equation that contains:

### Answer:

### Explanation:

A differential equation can contain derivatives, functions of the variable, and constants.

## 2. The order of the differential equation d^3y/dx^3 = x is:

### Answer:

### Explanation:

The order of a differential equation is determined by the highest order derivative present.

## 3. A differential equation of the form dy/dx = f(x) is called:

### Answer:

### Explanation:

It contains derivatives with respect to one variable only, hence ordinary.

## 4. The general solution of dy/dx = 0 is:

### Answer:

### Explanation:

The rate of change of y with respect to x is zero, which means y is a constant.

## 5. The solution of the differential equation d^2y/dx^2 = 0 is:

### Answer:

### Explanation:

Integrating once gives dy/dx = C1 and integrating again gives y = C1x + C2.

## 6. Which method is used to solve dy/dx = y/x?

### Answer:

### Explanation:

The variables can be separated to integrate each side individually.

## 7. The integrating factor of the differential equation dy/dx + Py = Q is:

### Answer:

### Explanation:

The integrating factor is given by the exponential of the integral of P with respect to x.

## 8. A first-order, first-degree differential equation is solvable by the method of separable variables if it can be expressed in the form:

### Answer:

### Explanation:

When it can be expressed as a product of functions of y and x, the variables can be separated.

## 9. The differential equation representing the family of circles x^2 + y^2 = a^2 is of order:

### Answer:

### Explanation:

We need one arbitrary constant (a) to represent this family, so the order is 1.

## 10. The linear differential equation of the first order is of the form:

### Answer:

### Explanation:

It's a first-order equation where y and its first derivative are in the first degree.

## 11. A homogeneous differential equation is one which:

### Answer:

### Explanation:

The term "homogeneous" refers to the degree of terms in the equation being consistent.

## 12. The solution of dy/dx = y/x with the condition y(1) = 1 is:

### Answer:

### Explanation:

Using separation of variables, we can integrate and apply the boundary condition to find the solution.

## 13. The differential equation representing the family of lines mx + y = 0 is of order:

### Answer:

### Explanation:

We only have one arbitrary constant (m), which means it's a zeroth-order differential equation.

## 14. A second order differential equation involves:

### Answer:

### Explanation:

It is based on the highest order of the derivative present.

## 15. The general solution of d^2y/dx^2 = 9y is:

### Answer:

### Explanation:

The given differential equation has constant coefficients, and its characteristic equation gives two real and distinct roots.

## 16. The differential equation whose solution is y = ae^x + be^-x is of order:

### Answer:

### Explanation:

There are two arbitrary constants a and b, so it's a second-order differential equation.

## 17. If y = e^(mx) is a solution of the differential equation a d^2y/dx^2 + b dy/dx + cy = 0, then m satisfies:

### Answer:

### Explanation:

Plugging y = e^(mx) into the differential equation yields the characteristic equation.

## 18. The solution of the differential equation dy/dx = y^2 is:

### Answer:

### Explanation:

Using separation of variables and integrating, we arrive at the solution.

## 19. The differential equation of all non-vertical lines in a plane is:

### Answer:

### Explanation:

The second derivative is zero for lines, indicating they have constant slope.

## 20. To find the particular solution of a differential equation, one needs:

### Answer:

### Explanation:

A particular solution is derived from the general solution by applying a specific condition.