Which of the following functions is a solution for the differential equation xy’ - y = 0?a)y = 4xb)y = x2c)y = -4xd)y = 2xCorrect answer is option 'D'. Can you explain this answer?

Question

Which of the following functions is a solution for the differential equation xy&rsquo; - y = 0?a)y = 4xb)y = x2c)y = -4xd)y = 2xCorrect answer is option 'D'. Can you explain this answer?

Answer

Consider the function y = 2x
Differentiating w.r.t x, we get
y’= dy/dx = 2
Substituting in the equation xy’-y, we get
xy’- y = x(2) - 2x = 2x - 2x = 0
Therefore, the function y = 2x is a solution for the differential equation xy’ - y = 0.

Answer

Given Differential Equation:
xy' - y = 0

Solution:

Step 1:
Let's first rewrite the given differential equation in standard form:
y' = y/x

Step 2:
To solve the differential equation, we can separate variables and integrate both sides:
dy/y = dx/x
∫(1/y)dy = ∫(1/x)dx
ln|y| = ln|x| + C
y = Cx

Step 3:
Now, we need to find the specific solution that satisfies the initial condition. In this case, we are given y = 2x as a potential solution.

Step 4:
Substitute y = 2x into the general solution:
2x = Cx
C = 2

Step 5:
Therefore, the specific solution for the differential equation xy' - y = 0 is:
y = 2x

Conclusion:
The correct solution for the given differential equation is y = 2x. So, option D, y = 2x, is the correct answer.

Top Courses for Class 12

Suggested Free Tests

Related Content

Schools

Competitive Examinations

Quick Access Links

Technical Exams