Differential Equations

Step 1: Rearranging.
We have: Multiply both sides by :

Step 2: Recognizing derivative form.
This suggests:

Step 3: Integration.
Integrating both sides w.r.t. :

Step 4: Simplify.

Final Answer:

Step 1: Identify the highest order derivative. In this equation, the highest order derivative is , which is the second-order derivative.

Step 2: Make sure the equation is polynomial in the derivatives. We need to ensure that the equation does not contain fractional or irrational powers of derivatives. In the given equation, is a polynomial expression in , and there are no fractional or irrational powers of the derivative.

Step 3: Identify the degree of the highest order derivative. The term involves squaring the second derivative, so the degree of the highest order derivative is 2. Conclusion: The degree of the given differential equation is .

Step 1: Rearrange the equation.
We can rearrange the equation as:

Step 2: Identify the type of equation.
This is a first-order linear differential equation with a non-constant coefficient. It is also a homogeneous equation because the right-hand side is a linear combination of and , and both terms on the right-hand side have degree one in and .

Step 3: Homogeneous and power one.
The equation is homogeneous because both the terms and appear to the first power, and there is no constant term that does not depend on or . It is also of power one because the degree of both and is one.

Step 4: Conclusion.
Thus, the correct answer is (D) Homogeneous and power one.

Step 1: Rearrange.

Step 2: Substitution. Let

Step 3: Rewrite. Substitute into original equation: Cancel :

Step 4: Rearrange terms.

Step 5: Solve linear ODE. This is linear in : Integrating factor:

Step 6: Integrate. Integration by parts: So,

Step 7: Back substitution.

Step 1: Differentiate once.

Step 2: Differentiate again.

Step 3: Compare with original function.

Final Answer:

Step 1: Rewrite equation. Take exponential both sides:

Step 2: Separate variables.

Step 3: Integrate both sides.

Step 4: Simplify. (where is constant of integration).

Final Answer:

Step 1: Definition of order of a differential equation.
The order of a differential equation is defined as the highest power of the derivative that appears in the equation.

Step 2: Analyze the given equation.
The given differential equation is: Here, the highest derivative is , which is the second derivative.

Step 3: Conclusion.
Since the highest derivative is the second derivative, the order of the differential equation is 2.

Final Answer: The correct answer is (C) .

Step 1: Given family of curves.

Step 2: Differentiate w.r.t. .

Step 3: Substitute in original equation.

Step 4: Eliminate . From the given: Differentiate w.r.t. : Which we already had. To eliminate , differentiate again.

Step 5: Differentiate again. Now eliminate using . Divide by 2:

Final Answer:

Step 1: Let

Step 2: Differentiate once.

Step 3: Differentiate again. Apply product rule: Now, So,

Step 4: Multiply through by . Now substitute :

Step 5: Rearranging.

Final Answer:

Step 1: Recall definition of order.
The order of a differential equation is the order of the highest derivative present in the equation.

Step 2: Identify highest derivative.
In the given equation: The highest derivative is .

Step 3: Ignore powers for order.
Even though is squared, the order depends only on the derivative itself, not its power.

Step 4: Conclusion.
Hence, the order of the differential equation is . The correct answer is (B).

Step 1: Standard form. This is a linear differential equation of the form:

Step 2: Find the integrating factor (IF).

Step 3: Multiply through by IF.

Step 4: Integrate both sides. Now,

Step 5: Final solution.

Final Answer:

Take the logarithm of both sides:

Differentiate with respect to :

Multiply through by :