If the function x, x2,, x3 are independent then the equation of differential equation formed with these independent solution x3y”’ – 3x2y” + 6xy’ – 6y equals to_______Correct answer is '0'. Can you explain this answer?

Question

If the function x, x2,, x3 are independent then the equation of differential equation formed with these independent solution x3y&#8221;&#8217; &#8211; 3x2y&#8221; + 6xy&#8217; &#8211; 6y equals to_______Correct answer is '0'. Can you explain this answer?

Answer

Understanding the Given Information

The given information states that the functions x, x^2, and x^3 are independent. This means that none of these functions can be written as a linear combination of the others. In other words, there is no constant value such that one of these functions can be obtained by multiplying another function by that constant.

Forming the Differential Equation

To form the differential equation using the given solutions, we can start by assuming that the dependent variable is y. Then, we can express the given solutions in terms of y as follows:
x^3y = y^4
3x^2y = 3xy^3
6xy = 6xy
6y = 6y

Combining the Expressions

Next, we can substitute these expressions into the given equation:
y^4 + 3xy^3 + 6xy + 6y = 0

Simplifying the Equation

To simplify the equation further, we can factor out a common factor of y:
y(y^3 + 3xy^2 + 6x + 6) = 0

Conclusion

From the simplified equation, we can see that there are two possible solutions for y: y = 0 and y^3 + 3xy^2 + 6x + 6 = 0. However, since the given information states that x, x^2, and x^3 are independent solutions, the only valid solution is y = 0.

Therefore, the correct answer to the differential equation formed with the independent solutions x, x^2, and x^3 is 0.

Similar Physics Doubts

Which of the following function form the solution of the differential equation,[x2D2–xD+ 2]y= x logx ?

If the differential equation has a solution y(x) = a cos (x) + b sin (x) for some non- zero real numbers a, b, then the value of must be _____ Correct answer is '5'. Can you explain this answer?

The differential equation dy/dx= 0.25 y² is to be solved using the backward (implicit) Euler's method with the boundary condition y = 1 at x = 0 and with a step size of 1. What would be the value of y at x = 1?(

The IF for the Differential Equation (1 + xy)y + x (1 - xy)dy/dx = 0 is xαyβ. Then find the value of theα +β. Correct answer is '-4'. Can you explain this answer?

Let u1, ... un be a linearly dependent set of functions on a x b, and let each function be(n 1) times differentiable in (a, b). Then

Top Courses for Physics

Suggested Free Tests

Related Content

Schools

Competitive Examinations

Quick Access Links

Technical Exams