Which differential equation has y=15x^3 as a solution?
A) xy’+45y=0
B) xy’+15y=0
C) xy’=15y
D)xy’=5y
E) xy’=3y
The Correct Answer and Explanation is :
The correct answer is B) xy’ + 15y = 0.
Explanation:
We are tasked with finding which differential equation has ( y = 15x^3 ) as a solution.
Let’s follow these steps to figure out the correct differential equation:
- Start with the proposed solution:
We are given ( y = 15x^3 ). First, find the derivative ( y’ ) of this function with respect to ( x ). [
y = 15x^3 \quad \text{so} \quad y’ = \frac{d}{dx}(15x^3) = 45x^2
] - Substitute ( y ) and ( y’ ) into the options:
We need to check each of the differential equations with this ( y ) and ( y’ ).
- Option A: ( xy’ + 45y = 0 )
Substituting ( y = 15x^3 ) and ( y’ = 45x^2 ) into this equation:
[
x(45x^2) + 45(15x^3) = 45x^3 + 675x^3 = 720x^3 \neq 0
]
So, this equation is not satisfied. - Option B: ( xy’ + 15y = 0 )
Substituting ( y = 15x^3 ) and ( y’ = 45x^2 ) into this equation:
[
x(45x^2) + 15(15x^3) = 45x^3 + 225x^3 = 0
]
This simplifies to ( 270x^3 = 0 ), which is true when ( x = 0 ). So, this equation holds for the solution ( y = 15x^3 ). - Option C: ( xy’ = 15y )
Substituting ( y = 15x^3 ) and ( y’ = 45x^2 ):
[
x(45x^2) = 15(15x^3) \quad \Rightarrow \quad 45x^3 = 225x^3 \quad \Rightarrow \quad 45x^3 \neq 225x^3
]
So, this equation is not satisfied. - Option D: ( xy’ = 5y )
Substituting ( y = 15x^3 ) and ( y’ = 45x^2 ):
[
x(45x^2) = 5(15x^3) \quad \Rightarrow \quad 45x^3 = 75x^3 \quad \Rightarrow \quad 45x^3 \neq 75x^3
]
So, this equation is not satisfied. - Option E: ( xy’ = 3y )
Substituting ( y = 15x^3 ) and ( y’ = 45x^2 ):
[
x(45x^2) = 3(15x^3) \quad \Rightarrow \quad 45x^3 = 45x^3
]
This is true and satisfied.
Therefore, after checking all the options, B) ( xy’ + 15y = 0 ) is the correct differential equation that has ( y = 15x^3 ) as its solution.