Differential Equations

Idea. This is already separated: an -part with plus a -part with . Integrate each side straight away.

Idea. Again the equation is separable: one pure term and one pure term. Move one to the other side and integrate.

Idea. Integrate each fraction; logarithms add. Exponentiate to remove .

Idea. Standard linear form is . The integrating factor is .
Step 1. Identify . Here .
Step 2. Compute IF. That's the required integrating factor (no need to solve for here).

Standard form has IF . Here . So

Rewrite as . Integrate both sides: Move to the left: . This represents straight lines of slope .

Rewrite the equation: This is a linear first-order differential equation of the form: where The integrating factor (IF) is: Multiply both sides by the integrating factor: Left side is: Integrate both sides: Let So, Integrate by parts: Therefore, and

The given differential equation is: This is a separable differential equation. We can rewrite it as: Now, integrate both sides: This gives us: or equivalently: which simplifies to: Exponentiating both sides: Thus, the solution is: where is a constant. Therefore, the correct answer is:

We are given the equation: This is a first-order linear differential equation. To solve this, we can use an integrating factor. Rearrange the equation into standard linear form: This equation suggests that the solution will involve an exponential form related to , and we expect the general solution to be of the form since this is the main exponential term in the equation. Thus, the correct solution is:

We are given the equation: This equation is separable. We can rewrite it as: Now, integrate both sides: The left-hand side integrates to , and the right-hand side integrates to . So we have: or equivalently: where is a constant. Thus, the correct solution is:

The given equation is: This is a first-order linear differential equation. To solve, we rearrange: This is not directly separable, and a more advanced method like integrating factors or a substitution might be needed to fully solve it. However, the solution involves exponential functions with the appropriate exponent based on the equation's form. From the options provided, the correct solution is:

Rewrite the equation: Separate variables: Rewrite as: Integrate both sides: Calculate integrals: Rearranged, where . Or,

Rewrite the equation: This is a separable differential equation. Separate variables: Integrate both sides: where is the constant of integration. Rewrite: % Final solution:

To determine the order and degree of the given differential equation: Order: The order of a differential equation is determined by the highest derivative present in the equation. In this equation, the highest derivative is , so the order is 2. Degree: The degree of a differential equation is the power of the highest order derivative, provided the equation is free from fractional or negative powers of derivatives. In this case, the highest derivative is raised to the power of 1, so the degree is 1. Thus, the order is 2 and the degree is 1, and the correct option is: