Differential Equations

We are given the differential equation: Step 1: Identifying the Order The order of a differential equation is the highest derivative present in the equation. From the given equation, - The highest derivative present is . Step 2: Identifying the Degree The degree of a differential equation is the exponent of the highest derivative after removing radicals and fractional powers. In the given equation, the second-order derivative appears as . To remove the square root (which is a fractional power), square both sides: Since the highest derivative term now appears with an exponent of 1, the degree is: Step 3: Summing Order and Degree Step 4: Correct Answer: (1) \ 5

Step 1: Rewrite the Differential Equation Given: Rewriting in standard form: Factor out : This is a separable differential equation.

Step 2: Separate the Variables Rearrange: Use partial fraction decomposition for the left-hand side. Let: Multiplying both sides by , we get: Substituting , we get . For , we get . Thus, rewriting:

Step 3: Integrate Both Sides Integrating separately:

Step 4: Simplify the Expression Taking exponentials: Rearrange: Solving for :

Step 5: Apply Initial Condition Substituting , : Solving for , we find . Thus, the final solution is:

(1) All circles passing through the origin. The general equation of a circle passing through the origin is There are two arbitrary constants and , so we can form a differential equation of order two.
(2) All parabolas passing through the origin and having focus on x-axis. The equation of such parabolas is . There is only one arbitrary constant , so we can form a differential equation of order one.
(3) All the lines passing through the origin. The equation of such lines is . There is only one arbitrary constant , so we can form a differential equation of order one.
(4) All hyperbolas of the form . There is only one arbitrary constant , so we can form a differential equation of order one. Thus, the only option that can form a differential equation of order two is (1).

Step 1: Identifying Order The order of a differential equation is the highest derivative present. In the given equation, the highest order derivative is , so the order is .

Step 2: Identifying Degree The degree of a differential equation is the power of the highest-order derivative after making the equation polynomial in derivatives. Since the equation contains negative and fractional powers of derivatives, we first remove these. These are non-polynomial terms, so we rewrite them appropriately and determine that the degree of the equation is .

Step 3: Compute the Difference