If Dy/Dx =6x^2 + 15x^4 and. Y=7 when x=2 ,find Y
dy/dx = 6 x^2 + 15 x^4
then
y = 2 x^3 + 3 x^5 + c
find c
7 = 2(8) + 3(32) + c
7 = 16 + 96 + c
c = -105
so
y = 2 x^3 + 3 x^5 - 105
Well, let's plug in the values given and have some fun with math!
If we have dy/dx = 6x^2 + 15x^4, we need to integrate it to find y. So, let's integrate it:
∫(dy/dx) dx = ∫(6x^2 + 15x^4) dx
Integrating the right side will give us:
∫(dy/dx) dx = 2x^3 + x^5 + C
Now, let's solve for y by substituting the given value:
7 = 2(2)^3 + (2)^5 + C
Simplifying:
7 = 16 + 32 + C
C = -41
Therefore, substituting C back into the equation, we have:
y = 2x^3 + x^5 - 41
So, when x = 2, y will be:
y = 2(2)^3 + (2)^5 - 41
= 16 + 32 - 41
= 48 - 41
= 7
And that's how we get y = 7 when x = 2. Keep smiling, my friend!
To find the value of Y, we need to solve the given differential equation and apply the initial condition.
Given: dy/dx = 6x^2 + 15x^4, and y = 7 when x = 2.
To integrate the differential equation, we integrate both sides with respect to x:
∫dy = ∫(6x^2 + 15x^4) dx
Integrating, we get:
y = 2x^3 + 3x^5 + C
Now, we need to apply the initial condition, y = 7 when x = 2:
7 = 2(2^3) + 3(2^5) + C
Simplifying:
7 = 16 + 96 + C
7 = 112 + C
C = 7 - 112
C = -105
Finally, we substitute C back into the equation for y:
y = 2x^3 + 3x^5 - 105
Therefore, when x = 2, y = 2(2^3) + 3(2^5) - 105 = 16 + 96 - 105 = 7.
So, when x = 2, y = 7.
To find the value of y, we need to integrate the given differential equation (dy/dx) and apply the initial condition, where y = 7 when x = 2.
Let's begin by integrating the given equation:
∫(dy/dx) dx = ∫(6x^2 + 15x^4) dx
Integrating the terms on the right-hand side separately, we get:
∫(dy/dx) dx = ∫6x^2 dx + ∫15x^4 dx
= 2x^3 + 3x^5 + C1
Where C1 is the constant of integration.
Now, by integrating the left-hand side and substituting y = 7 (when x = 2), we can find the value of C1.
∫dy = ∫(2x^3 + 3x^5 + C1) dx
y = (2/4)x^4 + (3/6)x^6 + C1x + C2
Since y = 7 when x = 2, we can substitute these values into the equation and solve for C2:
7 = (2/4)(2)^4 + (3/6)(2)^6 + C1(2) + C2
7 = 2 + 64/6 + 2C1 + C2
7 = 21⅓ + 2C1 + C2
Simplifying further:
7 = 63/3 + 2C1 + C2
7 = 21 + 2C1 + C2
7 - 21 = 2C1 + C2
-14 = 2C1 + C2
Now, to find the value of y, substitute the values of C1 and C2 into the equation:
y = (2/4)x^4 + (3/6)x^6 + C1x + C2
= (2/4)x^4 + (3/6)x^6 + (-7/2)x + 14
Hence, the value of y when x = 2 is:
y = (2/4)(2)^4 + (3/6)(2)^6 + (-7/2)(2) + 14
= 2 + 16 + (-7) + 14
= 25
Therefore, when x = 2, y is equal to 25.