Solution Of A Linear Equation

Step 1: Given differential equation
We are given: Rearranging, Step 2: Separating variables
Rewriting the equation: Dividing both sides by : Step 3: Integrating both sides
Integrating: Breaking into two integrals: Step 4: Evaluating the integrals
1. .
2. .
3. (since is treated as a constant).
Thus, Step 5: Conclusion
Thus, the correct answer is:

We are given the differential equation: Let , so . Now, differentiate with respect to : Substitute this into the original equation: Simplifying: Subtract from both sides: and divide both sides by : Now, separate variables: Integrating both sides: The integral of is , and the integral of is : Exponentiating both sides: Finally, substitute : Thus, the correct answer is .

Step 1: Rewrite the equation using the Pythagorean identity . Step 2: Solve the quadratic equation . We will solve it using the quadratic formula. The equation is in the form , where: Using the quadratic formula: Thus, the two possible values for are: Since is not a valid solution (because must lie between -1 and 1), we have: Step 3: Determine the values of that satisfy . The general solutions for are: Simplifying: This can be written as: Step 4: Conclusion Thus, the general solution is: