1What is the auxiliary equation for the differential equation ?
A.
B.
C.
D.
Correct Answer:
Explanation:To find the auxiliary equation (or characteristic equation), replace with . Thus, becomes .
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2If the roots of the auxiliary equation are real and distinct, say and , what is the complementary function (C.F.)?
A.
B.
C.
D.
Correct Answer:
Explanation:For distinct real roots and , the general solution for the homogeneous part is a linear combination of exponentials: .
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3Find the Complementary Function of .
A.
B.
C.
D.
Correct Answer:
Explanation:The auxiliary equation is , which simplifies to . The roots are (real and repeated). The C.F. is .
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4What is the Particular Integral (P.I.) of ?
A.
B.
C.
D.
Correct Answer:
Explanation:Using the operator method: . Since substituting gives 0, and the factor is squared, we use the formula . Here , so .
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5In the method of undetermined coefficients, if and 0 is not a root of the auxiliary equation, what is the correct form of the trial solution ?
A.
B.
C.
D.
Correct Answer:
Explanation:Since the right-hand side is a polynomial of degree 2, and 0 is not a root of the auxiliary equation, the trial solution is a general polynomial of degree 2: .
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6For the differential equation , what is the Particular Integral?
A.
B.
C.
D.
Correct Answer:
Explanation:We have where . Since makes the denominator zero, we use the case failure formula: .
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7Find the Wronskian of the functions and .
A.
B.$0$
C.$2$
D.
Correct Answer:
Explanation:.
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8In the method of variation of parameters, the particular solution is given by . What is the formula for ?
A.
B.
C.
D.
Correct Answer:
Explanation:By Cramer's rule applied to the variation of parameters derivation, , where is the non-homogeneous term and is the Wronskian.
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9Solve for P.I.: .
A.
B.
C.
D.
Correct Answer:
Explanation:.
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10Which substitution is used to transform a Cauchy-Euler homogeneous linear equation into a linear equation with constant coefficients?
A.
B.
C.
D.
Correct Answer:
Explanation:The standard substitution for Cauchy-Euler equations is or . This transforms terms like into constant coefficient operators.
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11Under the substitution , the operator transforms to which of the following (where )?
A.
B.
C.
D.
Correct Answer:
Explanation:Let . Then . Therefore, . So the operator transforms to .
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12Under the substitution , the term corresponds to which operator in terms of (where )?
A.
B.
C.
D.
Correct Answer:
Explanation:Using the chain rule repeatedly with , the term transforms to .
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13What is the P.I. of ?
A.
B.
C.
D.
Correct Answer:
Explanation:Substitute (the coefficient of in the exponent) into the operator function. .
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14If roots of the auxiliary equation are complex, , the Complementary Function is:
A.
B.
C.
D.
Correct Answer:
Explanation:Complex conjugate roots generate damped oscillatory solutions of the form .
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15Evaluate .
A.
B.
C.
D.
Correct Answer:
Explanation:The operator represents integration. is double integration. . .
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16What is the correct form of the particular solution for using undetermined coefficients?
A.
B.
C.
D.
Correct Answer:
Explanation:The auxiliary equation is , so roots are $1, 1$. Since is already in the C.F. twice (as and ), we must multiply the trial function by . So, .
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17For the differential equation , the Particular Integral is given by:
A.
B.
C.
D.
Correct Answer:
Explanation:This is a case of failure where . The formula is .
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18To solve simultaneous differential equations and , which method is most commonly used?
A.Method of Variation of Parameters
B.Operator method (Elimination)
C.Method of Undetermined Coefficients
D.Newton-Raphson Method
Correct Answer: Operator method (Elimination)
Explanation:The Operator method involves writing the system in terms of (where ) and eliminating one variable to solve for the other, similar to solving simultaneous algebraic equations.
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19Calculate .
A.
B.
C.
D.
Correct Answer:
Explanation:Replace with . .
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20What is the value of where is a function of ?
A.
B.
C.
D.
Correct Answer:
Explanation:This is the Exponential Shift Theorem. When an exponential factor multiplies a function , we can pull to the left of the operator by replacing with .
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21Which differential equation is a Legendre's Linear Equation (a generalized form of Euler-Cauchy)?
A.
B.
C.
D.
Correct Answer:
Explanation:Legendre's linear equation involves terms of the form . The Euler-Cauchy equation is a specific case where .
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22Solve .
A.
B.
C.
D.
Correct Answer:
Explanation:Auxiliary equation: . Roots are . Solution is .
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23In the method of undetermined coefficients, if and roots of aux eq are , the trial function is:
A.
B.
C.
D.
Correct Answer:
Explanation:Since corresponds to roots , and these roots match the complementary function roots, we must multiply the standard trial form () by .
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24Find .
A.
B.
C.
D.
Correct Answer:
Explanation:Substitute . .
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25In the method of variation of parameters for a second order DE, how many arbitrary constants are replaced by functions of ?
A.1
B.2
C.3
D.
Correct Answer: 2
Explanation:For a second-order linear differential equation, the method involves replacing the two arbitrary constants and from the complementary function with unknown functions and .
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26What is the P.I. for ?
A.
B.
C.
D.
Correct Answer:
Explanation:Substitute . .
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27Transform the Euler-Cauchy equation into a linear DE with constant coefficients using .
A.
B.
C.
D.
Correct Answer:
Explanation:Substitute terms: and . The equation becomes .
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28What is ?
A.
B.
C.
D.Undefined
Correct Answer:
Explanation:Using the specific formula for the resonant case: .
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29For the system and , what is the resulting differential equation for after eliminating ?
30In the Variation of Parameters method, the condition imposed on and to simplify the derivation is:
A.
B.
C.
D.
Correct Answer:
Explanation:To ensure we can solve for and , we impose the condition so that the first derivative of the trial solution does not contain derivatives of the parameters.
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31Evaluate .
A.
B.
C.
D.Undefined
Correct Answer:
Explanation:Substitute . .
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32When solving simultaneous differential equations, if the roots of the auxiliary equation for are , what is the form of ?
A.
B.
C.
D.
Correct Answer:
Explanation:Roots correspond to purely imaginary roots with . The solution is .
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33Calculate .
A.
B.
C.
D.
Correct Answer:
Explanation:Use shift theorem: .
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34Identify the non-homogeneous linear differential equation.
A.
B.
C.
D.
Correct Answer:
Explanation:A non-homogeneous linear DE has a non-zero term containing only the independent variable (or constants) on the RHS, and the dependent variable and its derivatives appear to the first power. Option B fits. Option C and D are non-linear.
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35What is the P.I. of ?
A.$2$
B.$1$
C.$0$
D.
Correct Answer: $2$
Explanation:The RHS is a constant, which can be written as . Substitute into . .
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36Which method is generally capable of solving linear differential equations with variable coefficients?
A.Undetermined Coefficients
B.Operator Method (Shortcut)
C.Variation of Parameters
D.Partial Fractions
Correct Answer: Variation of Parameters
Explanation:The method of Variation of Parameters is powerful and can be applied to linear equations with variable coefficients, provided the fundamental set of solutions (C.F.) is known.
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37For , what are the roots of the auxiliary equation after substitution ?
A.
B.$1, 1$
C.
D.$0, 1$
Correct Answer:
Explanation:Substitute to get . Roots are .
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38Find the general solution of .
A.
B.
C.
D.
Correct Answer:
Explanation:Aux eq: . Real distinct roots.
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39Evaluate .
A.
B.
C.
D.
Correct Answer:
Explanation:Denominator is . We have . This is a case of failure (twice). Formula: .
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40If for all , then and are:
A.Linearly Independent
B.Linearly Dependent
C.Orthogonal
D.Constants
Correct Answer: Linearly Dependent
Explanation:A zero Wronskian implies that the functions are linearly dependent.
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41Find .
A.
B.
C.
D.
Correct Answer:
Explanation:. .
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42What is the P.I. of ?
A.
B.
C.
D.
Correct Answer:
Explanation:Substitute . .
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43For the equation , what substitution transforms it to constant coefficients?
A.
B.
C.
D.
Correct Answer:
Explanation:This is a 3rd order Euler-Cauchy equation. The same substitution applies to all orders.
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44When finding P.I. for , one should treat as:
A.
B.
C.
D.Polynomial
Correct Answer:
Explanation:It is easiest to express hyperbolic functions in terms of exponentials and use the rule for each term.
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45Solve .
A.
B.
C.
D.
Correct Answer:
Explanation:Aux eq: . Solution .
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46The trial solution for undetermined coefficients for would be:
A.
B.
C.
D.
Correct Answer:
Explanation:By the principle of superposition, we sum the trial solutions for the polynomial part () and the exponential part ().
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47What is the P.I. of ?
A.
B.
C.
D.
Correct Answer:
Explanation:.
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48If the auxiliary equation has roots $0, 0, 2$, the C.F. is:
A.
B.
C.
D.
Correct Answer:
Explanation:The root 0 is repeated twice, contributing . The root 2 contributes .
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49In simultaneous DEs, if and , find .
A.
B.
C.
D.
Correct Answer:
Explanation:Integrate with respect to . .
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50Which property allows us to write the general solution as ?
A.Linearity
B.Homogeneity
C.Continuity
D.Differentiability
Correct Answer: Linearity
Explanation:For linear differential equations, the general solution is the sum of the complementary function (solution to homogeneous eq) and the particular integral (specific solution to non-homogeneous eq).
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