Differential Equations

Step 1: Understanding the differential equation.
The given differential equation is: This is a second-order linear homogeneous differential equation with constant coefficients. The general form of the solution to this equation is: This solution comes from the fact that the characteristic equation for this differential equation is , which has the roots , leading to a solution involving sine and cosine functions.

Step 2: Applying the initial conditions.
We are given the initial conditions: 1. 2. Let’s first apply the initial condition : Substitute into the general solution: Since , we get: Now, apply the second initial condition : Substitute into the general solution: Since and , this simplifies to: Since , we have:
Step 3: Final solution.
Substitute the values of and into the general solution: Thus, the solution is:

Step 1: Understand the differential equation.
The given differential equation is: This is a first-order nonlinear differential equation. To solve it, we will use a method called separation of variables. However, the equation is already suggestive of a transformation that can simplify it. We will introduce a substitution to simplify the equation further.
Step 2: Apply substitution.
Let’s introduce a new variable . This substitution simplifies the terms, and we can now express as . Differentiating this expression with respect to , we get: Now, substitute this expression for into the original equation: Simplify:
Step 3: Rearrange and simplify.
Canceling out the terms from both sides gives: Now, separate the variables so that all -terms are on one side and all -terms are on the other side:
Step 4: Integrate both sides.
Now, integrate both sides: The integral on the left side is , and the integral on the right side is . Thus, we have: Simplifying, we get: We can replace with another constant, say , and rewrite the equation as: Thus, the general solution is: where is a constant.
Step 5: Conclusion.
The correct general solution to the differential equation is:

Step 1: Understanding the differential equation.
The given equation is , which is a first-order linear differential equation. To solve it, we will first separate the variables and integrate.

Step 2: Solving the differential equation.
Rearrange the terms to separate and . By simplifying and integrating both sides, we obtain the solution: This is the general solution to the differential equation.

Step 3: Conclusion.
The correct answer is (B) , as it is the solution to the given differential equation.

Step 1: Understanding the differential equation.
The given differential equation is: This is a first-order linear differential equation. We can rewrite it as: To solve this equation, we recognize that it is in the form of a linear differential equation, and we can use the method of integrating factor to solve it.
Step 2: Transforming the equation.
Rearrange the equation: This is now in the standard form of a linear differential equation: where and .
Step 3: Finding the integrating factor.
The integrating factor is given by:
Step 4: Multiplying by the integrating factor.
Multiply both sides of the differential equation by the integrating factor : This simplifies to: Now, we integrate both sides with respect to .
Step 5: Integrating both sides.
Integrating both sides: The left side becomes , and the right side integrates to : where is the constant of integration.
Step 6: Solving for .
Multiply both sides by to solve for :
Step 7: Applying the initial condition.
We are given the initial condition . Substitute and into the equation: Since , we get: Thus, .
Step 8: Final solution.
Substitute into the general solution: Thus, the solution to the differential equation is:
Step 9: Conclusion.
The correct solution is (D) .

Step 1: Understanding the differential equation.
We are given the second-order linear homogeneous differential equation: This is a linear differential equation with constant coefficients. To solve this type of equation, we first write down the characteristic equation and solve for the roots.
Step 2: Finding the characteristic equation.
Assume that the solution is of the form , where is a constant. Then, we can calculate the first and second derivatives: Substituting these into the given equation: Factor out (which is never zero): Thus, the characteristic equation is:
Step 3: Solving the characteristic equation.
Now, solve the quadratic equation . Factorizing: So, the roots are:
Step 4: General solution.
Since the roots and are real and distinct, the general solution to the differential equation is: Substituting the values of and :
Step 5: Conclusion.
The general solution to the given differential equation is: Therefore, the correct answer is: