Question:
Number of first order equations in a system equivalent to the equation \(x''' + 3x'' + 2x' - 5x = \sin 2t\) is:
Number of first order equations in a system equivalent to the equation \(x''' + 3x'' + 2x' - 5x = \sin 2t\) is:
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Exam Tip:
An \(n\)-th order ODE can always be written as a system of \(n\) first-order ODEs.
An \(n\)-th order ODE can always be written as a system of \(n\) first-order ODEs.
Updated On: Aug 27, 2026
- 1
- 2
- The given system is irreducible
- 3
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The Correct Option is D
Solution and Explanation
Step 1: Understanding the Concept:
A higher-order ordinary differential equation can be converted into a system of first-order equations by introducing new variables.
Step 2: Key Formula or Approach:
The order of the differential equation is the highest derivative present.
Here, the highest derivative is \(x'''\), which is the third derivative.
So, the order is 3.
A third-order ODE is equivalent to a system of three first-order ODEs.
Step 3: Detailed Explanation:
Let \(x_1 = x\), \(x_2 = x'\), \(x_3 = x''\).
Then, \[ x_1' = x' = x_2 \] \[ x_2' = x'' = x_3 \] \[ x_3' = x''' = -3x'' - 2x' + 5x + \sin 2t = -3x_3 - 2x_2 + 5x_1 + \sin 2t \] So, we have three first-order equations.
Thus, the number of first-order equations is 3.
Step 4: Final Answer:
Therefore, option (D) is correct.
A higher-order ordinary differential equation can be converted into a system of first-order equations by introducing new variables.
Step 2: Key Formula or Approach:
The order of the differential equation is the highest derivative present.
Here, the highest derivative is \(x'''\), which is the third derivative.
So, the order is 3.
A third-order ODE is equivalent to a system of three first-order ODEs.
Step 3: Detailed Explanation:
Let \(x_1 = x\), \(x_2 = x'\), \(x_3 = x''\).
Then, \[ x_1' = x' = x_2 \] \[ x_2' = x'' = x_3 \] \[ x_3' = x''' = -3x'' - 2x' + 5x + \sin 2t = -3x_3 - 2x_2 + 5x_1 + \sin 2t \] So, we have three first-order equations.
Thus, the number of first-order equations is 3.
Step 4: Final Answer:
Therefore, option (D) is correct.
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