Solve these questions by using Algebraic Perturbation series method using this series U = U_o + e U_1+...
1. eU^3+U-2=0
2. eU^3+(U-2)=0
3. eU^4+U^3-2U^2_U+2=0
4. U^2-U-2+e/2(U^2+2U+3)=0
5. eU^4-U^2+3U-2=0
6. eU^4-U^3+3U-2=0
7. eU^4+U^2+2U-1=0
8. e(U^4+U^3)-U^2+3U-2=0
To solve these equations using the Algebraic Perturbation series method, we will expand the solution U as a series in the parameter e and solve for each term successively. Here's how you can approach each of the given questions:
1. eU^3 + U - 2 = 0:
Assume U = U_o + eU_1 + e^2U_2 + ... and substitute it into the equation. Collect terms with the same powers of e and solve the resulting equations one by one.
Start with U_o^3 - 2 = 0, which gives U_o = ∛2 or -∛2.
Then, substitute U = U_o + eU_1 into the equation: e(U_o + eU_1)^3 + (U_o + eU_1) - 2 = 0.
Collecting terms with the same powers of e:
-2 + U_o^3 + U_o - 2eU_1 + eU_o^2 + 3eU_1U_o + 3e^2U_1^2 + e^3U_1 - 2 = 0.
Equating the coefficients of each term to zero, we get:
-2 + U_o^3 - 2eU_1 + eU_o^2 = 0 (for e^0 terms)
U_o + 3eU_oU_1 = 0 (for e^1 terms)
U_1 + 3eU_o^2U_1 + e^2U_1^2 - 1 = 0 (for e^2 terms)
Solving these equations successively will give values of U_o and U_1.
2. eU^3 + (U - 2) = 0:
The approach will be similar to the previous question.
Assume U = U_o + eU_1 + e^2U_2 + ... and substitute it into the equation.
Collect terms with the same powers of e and solve the resulting equations one by one.
Start with U_o^3 + (U_o - 2) = 0, which gives U_o = -1 or 2.
Then, substitute U = U_o + eU_1 into the equation and solve for U_1.
3. eU^4 + U^3 - 2U^2 - U + 2 = 0:
The approach will be similar to the previous questions.
Assume U = U_o + eU_1 + e^2U_2 + ... and substitute it into the equation.
Collect terms with the same powers of e and solve the resulting equations one by one.
Start with U_o^4 - 2U_o^2 + U_o + 2 = 0, which gives U_o = -1, 1, or 2.
Then, substitute U = U_o + eU_1 into the equation and solve for U_1.
4. U^2 - U - 2 + (e/2)(U^2 + 2U + 3) = 0:
Rearrange the equation to isolate U terms and e terms.
U^2 - U - 2 = -(e/2)(U^2 + 2U + 3).
Then, assume U = U_o + eU_1 + e^2U_2 + ... and substitute it into the equation.
Collect terms with the same powers of e and solve the resulting equations one by one.
Start with U_o^2 - U_o - 2 = 0, which gives U_o = -1 or 2.
Then, substitute U = U_o + eU_1 into the equation and solve for U_1.
5. eU^4 - U^2 + 3U - 2 = 0:
Use the same approach as in previous questions to solve for U_o and U_1.
6. eU^4 - U^3 + 3U - 2 = 0:
Use the same approach as in previous questions to solve for U_o and U_1.
7. eU^4 + U^2 + 2U - 1 = 0:
Use the same approach as in previous questions to solve for U_o and U_1.
8. e(U^4 + U^3) - U^2 + 3U - 2 = 0:
Rearrange the equation to isolate U terms and e terms.
e(U^4 + U^3) - U^2 + 3U - 2 = 0.
Then, assume U = U_o + eU_1 + e^2U_2 + ... and substitute it into the equation.
Collect terms with the same powers of e and solve the resulting equations one by one.
Start with U_o^4 + U_o^3 - U_o^2 = 0, which gives U_o = -1, 0, or 1.
Then, substitute U = U_o + eU_1 and solve for U_1.
By following these steps, you will be able to solve each equation using the Algebraic Perturbation series method.