By inspection, find a particular solution of y” + 2y = 14. Yp(x)-
The Correct Answer and Explanation is :
A particular solution ( Y_p(x) ) for the differential equation:
[
y” + 2y = 14
]
can be found by inspection.
Step 1: Identify the Non-Homogeneous Term
The right-hand side of the equation is a constant ( 14 ). This suggests trying a constant solution for ( Y_p(x) ).
Step 2: Assume a Particular Solution Form
We assume a constant solution:
[
Y_p = C
]
where ( C ) is a constant to be determined.
Step 3: Compute Derivatives
Since ( Y_p = C ) is a constant, its first and second derivatives are:
[
Y_p’ = 0, \quad Y_p” = 0.
]
Step 4: Substitute into the Differential Equation
Substituting into the given equation:
[
0 + 2(C) = 14.
]
Step 5: Solve for ( C )
[
2C = 14
]
[
C = 7.
]
Conclusion: Particular Solution
Thus, a particular solution to the equation is:
[
Y_p(x) = 7.
]
Explanation (300 words)
To find a particular solution of ( y” + 2y = 14 ), we analyze the non-homogeneous term. The function on the right-hand side is a constant, which suggests trying a constant function as a particular solution.
If ( y_p(x) = C ), then the derivatives are zero: ( y_p’ = 0 ) and ( y_p” = 0 ). Substituting into the equation:
[
0 + 2C = 14.
]
Solving for ( C ), we divide by 2 to get ( C = 7 ). This means the function ( Y_p(x) = 7 ) satisfies the equation.
This method works because when the non-homogeneous term is a constant, the differential equation effectively balances the constant with the coefficient of ( y ). Since the equation contains ( 2y ), solving for ( y ) directly gives ( y = 7 ).
A key point in this approach is recognizing the right-hand function’s form and choosing an appropriate trial solution. If the right-hand term were more complex (e.g., polynomial, exponential, or trigonometric), we would use different techniques like undetermined coefficients or variation of parameters.
Finding a particular solution is essential in solving non-homogeneous differential equations because it helps construct the general solution, which consists of the homogeneous solution plus this particular solution.
Now, let’s generate a visual representation of this process.
Here is a visual representation of the particular solution ( Y_p(x) = 7 ). The red horizontal line at ( y = 7 ) illustrates that the solution is constant for all values of ( x ), as expected from our calculations. Let me know if you need further clarification!