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Consider the differential equation 3yn(x) + 27y(x) = 0 with initial conditions y(0) = 0 and y'(0) = 2000. The value of y at x = 1 is
- a)93
- b)95
Correct answer is between '93,95'. Can you explain this answer?
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Verified Answer
Consider the differential equation 3yn(x) + 27y(x) = 0 with initial c...
3y′′(x) + 27y(x) = 0, y(0) = 0, y′(0) = 2000
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Auxiliary equation, 3m2 + 27 = 0 ⇒ m2 + 9 = 0
⇒ m = 0 + 3i
yc = c1 cos 3x + c2 sin3x and yp = 0
∴ yc = c1 cos 3x + c2 sin 3x
y(0) = 0 ⇒ c1 + 0 = 0 ⇒ c1 = 0
∴ y = c2 sin3x
y = 3c2 cos3x
y′(0) = 2000 ⇒ 2000 = 3c2 ⇒ c2 = 2000/3
Here in sin 3, 3 is in radian as it is a value.
Most Upvoted Answer
Consider the differential equation 3yn(x) + 27y(x) = 0 with initial c...
Understanding the Differential Equation
The given differential equation is:
3y''(x) + 27y(x) = 0
This can be simplified to:
y''(x) + 9y(x) = 0
This is a second-order linear homogeneous differential equation with constant coefficients.
Characteristic Equation
To solve this, we first find the characteristic equation:
m² + 9 = 0
This yields the roots:
m = ±3i
These roots indicate that the general solution will be a combination of sine and cosine functions:
General Solution
The general solution is:
y(x) = C₁ cos(3x) + C₂ sin(3x)
Applying Initial Conditions
Now, we apply the initial conditions:
1. y(0) = 0:
- From the general solution, y(0) = C₁ = 0
2. y'(0) = 2000:
- Differentiating y(x) gives:
- y'(x) = -3C₁ sin(3x) + 3C₂ cos(3x)
- Substituting initial conditions:
- y'(0) = 3C₂ = 2000 ⇒ C₂ = 666.67
Thus, the solution simplifies to:
y(x) = 666.67 sin(3x)
Calculating y(1)
Now, substitute x = 1 into the equation to find y(1):
y(1) = 666.67 sin(3)
Using the approximate value of sin(3) ≈ 0.1411:
y(1) ≈ 666.67 * 0.1411 ≈ 94.3
This value lies between 93 and 95.
Conclusion
Thus, the value of y at x = 1 is correctly approximated to be between 93 and 95.
The given differential equation is:
3y''(x) + 27y(x) = 0
This can be simplified to:
y''(x) + 9y(x) = 0
This is a second-order linear homogeneous differential equation with constant coefficients.
Characteristic Equation
To solve this, we first find the characteristic equation:
m² + 9 = 0
This yields the roots:
m = ±3i
These roots indicate that the general solution will be a combination of sine and cosine functions:
General Solution
The general solution is:
y(x) = C₁ cos(3x) + C₂ sin(3x)
Applying Initial Conditions
Now, we apply the initial conditions:
1. y(0) = 0:
- From the general solution, y(0) = C₁ = 0
2. y'(0) = 2000:
- Differentiating y(x) gives:
- y'(x) = -3C₁ sin(3x) + 3C₂ cos(3x)
- Substituting initial conditions:
- y'(0) = 3C₂ = 2000 ⇒ C₂ = 666.67
Thus, the solution simplifies to:
y(x) = 666.67 sin(3x)
Calculating y(1)
Now, substitute x = 1 into the equation to find y(1):
y(1) = 666.67 sin(3)
Using the approximate value of sin(3) ≈ 0.1411:
y(1) ≈ 666.67 * 0.1411 ≈ 94.3
This value lies between 93 and 95.
Conclusion
Thus, the value of y at x = 1 is correctly approximated to be between 93 and 95.
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Consider the differential equation 3yn(x) + 27y(x) = 0 with initial conditions y(0) = 0 and y'(0) = 2000. The value of y at x = 1 isa)93b)95Correct answer is between '93,95'. Can you explain this answer?
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