Differential Equations

We are given the differential equation and we need to find the general solution.

First, rewrite the equation as:

Divide both sides by :

This is a first-order linear differential equation in the form , where and .

Find the integrating factor .

Multiply the differential equation by the integrating factor:

Integrate both sides with respect to x:

Therefore, the general solution of the differential equation is .

Thus, the correct option is (A) .

Step 1: Express the equation in a more manageable form.

The equation can be rewritten as:

$ c_1 c_2 c_3 c_4 x x y \boxed{2} $.

We are given the differential equation: This is a separable differential equation. We can separate the variables: Now, integrate both sides: The integrals are straightforward: Exponentiating both sides gives: where . Now, solve for : Using the initial condition : This is a contradiction, indicating no solution exists for the given initial condition. Thus, the number of solutions is infinite for this differential equation because there is no unique solution that satisfies the initial condition.

The correct answer is both (B) : one and (C) : infinite.

The given differential equation is: Rearranging terms: This is a separable differential equation. By separating variables: Integrating both sides: This implies:

So, the correct answer is (C) : Straight line passing through origin.

To solve the differential equation , let's follow the steps:

Rearrange the terms to facilitate separation of variables:

Divide both sides by to separate the variables:

Integrate both sides:

The integrals yield:

where is the constant of integration.

Exponentiate both sides to eliminate the logarithm:

which gives:

Let (where is a positive constant), then:

This implies:

or , representing straight lines through the origin with slopes and respectively.

Thus, the solution of the differential equation indeed represents a straight line passing through the origin.

This justifies the correct answer as: Straight line passing through the origin.

Given:
Differential equation: where , and

Step 1: Identify the order
The highest derivative is , so:
Order = 2

Step 2: Identify the degree
Since the equation involves a fractional exponent, we must first remove the root by raising both sides to the power of 3:
Now the equation is polynomial in the highest derivative , and its highest power is 3, so:
Degree = 3

Sum of degree and order:

Answer: 5

We are given:
Let us use substitution. Let:
Since , we get: Now separate the variables: Integrate both sides: Recall , so: ✅ Correct answer:

To find the values of and , we can compare the given integral expression with the expression:

Comparing the integrand of the given integral with the expression above, we can see that:

Therefore, option (A) is the correct answer.

We are given:

Step 1: Differentiate implicitly with respect to
Group terms:

Step 2: Find value of at
From original equation: Substitute into :

Step 3: Differentiate again to find
Differentiate: Use quotient rule: At , compute numerator: Denominator: So:

Final answer: