Question

Which of the following is a solution to the differential equation xy′−3y=6 ?
y=x^2+2
y=3x^2−2
y=5x^3−2
y=7x^3+2

Answer 1

Answer:

$y = 5 {x}^{3} - 2$

Step-by-step explanation:

The given differential equation is :

xy′−3y=6

We want to determine which of the following options is a solution to the differential equations.

The function that satisfies the differential equation is a solution.

We can verify that

$y = 5 {x}^{3} - 2$

satisfy this differential equation.

We differentiate to get:

$y' = 15 {x}^{2}$

We substitute the function and its derivative into the differential equation to get:

$x(15 {x}^{2}) - 3(5 {x}^{3} - 2) = 6$

We expand and simplify on the left:

$15 {x}^{3}- 15 {x}^{3} + 6= 6$

This simplifies to:

$6 = 6$

Verified.

We can show that all the other functions do not satisfy this differential equation

Answer 2

The solution of the given differential equation is $y=5x^3-2$.

Given information:

The differential equation is $xy'-3y=6$.

It is required to find the solution of the differential equation.

So, rearrange the differential equation as,

$xy'-3y=6\\x\dfrac{dy}{dx}-3y=6$

By analysis, we can say that the equation $y=5x^3-2$ is the solution of the given differential equation because it satisfies the differential equation.

Check for the solution as,

$x\dfrac{dy}{dx}-3y=6\\x\dfrac{d(5x^3-2)}{dx}-3(5x^3-2)=6\\x(15x^2-0)-15x^3+6=6\\15x^3-15x^3+6=6\\6=6$

Hence proved.

Therefore, the solution of the given differential equation is $y=5x^3-2$.

For more details, refer to the link:

https://brainly.com/question/353770