1A differential equation of the form is said to be exact if:
A.
B.
C.
D.
Correct Answer:
Explanation:The necessary and sufficient condition for the differential equation to be exact is that the partial derivative of with respect to equals the partial derivative of with respect to .
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2If the equation is exact, the general solution is given by:
A.
B.
C.
D.
Correct Answer:
Explanation:The standard formula for solving an exact differential equation is to integrate with respect to (treating as constant) and integrate only those terms of that do not contain with respect to , then equate the sum to a constant.
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3Check if the equation is exact.
A.Yes, because
B.No, because
C.Yes, because
D.It is not a first-order equation.
Correct Answer: Yes, because
Explanation:Here and . and . Since they are equal, the equation is exact.
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4The solution to the exact equation is:
A.
B.
C.
D.
Correct Answer:
Explanation:The equation is exact (). Integrating gives . Alternatively, using the formula: .
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5If , a function of only, then the integrating factor (I.F.) is:
A.
B.
C.
D.
Correct Answer:
Explanation:This is a standard rule for finding an integrating factor. If the expression depends only on , the I.F. is .
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6For a homogeneous differential equation , if , then the integrating factor is:
A.
B.
C.
D.
Correct Answer:
Explanation:For a homogeneous equation where and are of the same degree, provided , the integrating factor is .
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7Find the integrating factor for the equation of the form , given .
A.
B.
C.
D.
Correct Answer:
Explanation:When the equation is of the form , the integrating factor is .
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8Find the value of such that is exact.
A.1
B.2
C.3
D.
Correct Answer: 1
Explanation:, . For exactness, . , . Thus, , which implies .
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9What is the integrating factor of the linear differential equation , where and are functions of ?
A.
B.
C.
D.
Correct Answer:
Explanation:This is the standard definition of the integrating factor for a first-order linear differential equation.
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10For the equation , determine the integrating factor treating it as a homogeneous equation.
A.
B.
C.
D.
Correct Answer:
Explanation:, . . (Wait, checking options/calculation). Let's re-evaluate standard I.F. options for Homogeneous. . . This is not listed simply. Let's check calculation again. , . . This simplifies to . If we check . The options provided seem to imply a specific result or a typo in the question context usually found in textbooks. However, looking at the problem as , usually homogeneous I.F. is . Let's verify if there is a simpler I.F. often used. Dividing by makes it exact? , . Often is an I.F. Let's calculate and for . , . . . Yes, works. The Homogeneous rule is sufficient, not necessary. Among options, is valid.
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11Solve the equation .
A.
B.
C.
D.
Correct Answer:
Explanation:, . , . It is exact. . Terms in free from : None. Thus .
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12If , a function of only, the integrating factor is:
A.
B.
C.
D.
Correct Answer:
Explanation:This is the rule for finding an integrating factor dependent only on .
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13In the differential equation , calculate .
A.
B.
C.
D.$0$
Correct Answer:
Explanation:, . . Wait, let me recompute. . . The option is correct. My previous calculation in option A was hasty.
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14The integrating factor for the equation is:
A.
B.
C.
D.
Correct Answer:
Explanation:Rewrite in standard linear form: . Here . I.F. = .
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15For a differential equation of first order and higher degree, let . If the equation is solvable for , it can be expressed as:
A.
B.
C.
D.
Correct Answer:
Explanation:If an equation is solvable for , it means can be expressed explicitly as a function of and .
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16Solve the equation where .
A.
B.
C.
D.
Correct Answer:
Explanation:Factor the quadratic: . Thus or . Integrating gives and . The combined general solution is .
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17A differential equation of the form is known as:
A.Bernoulli's Equation
B.Clairaut's Equation
C.Lagrange's Equation
D.Euler's Equation
Correct Answer: Clairaut's Equation
Explanation:The form is the standard form of Clairaut's equation.
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18What is the general solution of Clairaut's equation ?
A.
B.
C.
D.
Correct Answer:
Explanation:To find the general solution of Clairaut's equation, we simply replace with an arbitrary constant .
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19Which of the following represents the singular solution of a differential equation?
A.A solution containing arbitrary constants.
B.A particular case of the general solution.
C.A solution that contains no arbitrary constants and is not deducible from the general solution by giving values to constants.
D.A solution obtained by setting constant .
Correct Answer: A solution that contains no arbitrary constants and is not deducible from the general solution by giving values to constants.
Explanation:A singular solution is an envelope of the family of curves represented by the general solution and cannot be obtained by specific constant substitution.
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20Find the general solution of .
A.
B.
C.
D.
Correct Answer:
Explanation:This is Clairaut's form with . Replace with to get .
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21To solve an equation solvable for of the form , we differentiate with respect to:
A.
B.
C.
D. and
Correct Answer:
Explanation:When an equation is expressed as , the standard method is to differentiate with respect to , allowing us to use .
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22The singular solution of the equation is:
A.
B.
C.
D.
Correct Answer:
Explanation:General solution: . This is a quadratic in : . For singular solution, discriminant . Here .
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23Which equation is reducible to Clairaut's form?
A.
B.
C.
D.None of these
Correct Answer:
Explanation:Often equations involving can be manipulated or substituted to fit Clairaut's form. Specifically, algebraic manipulation is required. is Clairaut's.
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24If an equation is of the form , where , the solution is:
A.
B. where
C.
D.
Correct Answer: where
Explanation:Since must be a constant roots of , let root be . Then .
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25Find the integrating factor for using the rule for or similar.
A.
B.
C.
D.This problem requires specific rule checking (Rule: or similar)
Correct Answer:
Explanation:Let's check , . , . . . Divide by : . This is a function of . I.F. = .
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26The differential equation can be factored into:
A.
B.
C.
D.
Correct Answer:
Explanation:Expand options: . Matches the equation.
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27What is the general solution of ?
A.
B.
C.
D.
Correct Answer:
Explanation:It is a Clairaut's equation. Replace with .
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28The equation can be reduced to Clairaut's form using the substitution:
A.
B.
C.
D.
Correct Answer:
Explanation:This is a standard substitution for equations of this type to reduce to .
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29Determine if is exact.
A.Exact
B.Not Exact
C.Exact only if
D.Undefined
Correct Answer: Exact
Explanation:. . Since , it is exact.
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30Solve the exact equation .
A.
B.
C.
D.
Correct Answer:
Explanation:Integrate w.r.t : . Integrate terms of (free of ) w.r.t : . Sum: .
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31For the equation , solve for .
A.
B.
C.
D.Cannot be solved for
Correct Answer:
Explanation:Rearranging the terms: .
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32The term 'degree' in 'Equations of the first order and higher degree' refers to the power of:
A.
B.
C.
D.The highest derivative present
Correct Answer:
Explanation:The degree of a differential equation is the power of the highest derivative (here first order ) after the equation has been made rational and integral in derivatives.
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33Find the general solution of the differential equation .
A.
B.
C.
D.
Correct Answer:
Explanation:.
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34Which of the following is an Integrating Factor for ?
A.
B.
C.
D.Both A and C are correct
Correct Answer: Both A and C are correct
Explanation: becomes with and logic with . Both make the equation integrable. In multiple choice, usually is the primary textbook answer for reducibility to derivative of quotient. Let's check exactness. . 1) I.F : . . Exact. 2) I.F : . . Exact. Both are valid. We select the option stating 'Both'.
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35Identify the type of equation: .
A.Linear Equation
B.Bernoulli's Equation
C.Clairaut's Equation
D.Exact Equation
Correct Answer: Clairaut's Equation
Explanation:It matches the form where .
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36If , find the integrating factor.
A.
B.
C.
D.
Correct Answer:
Explanation:This is a function of (constant 2 is independent of ). I.F. = .
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37The envelope of the family of lines represents:
A.The general solution
B.The singular solution
C.The particular solution
D.None of the above
Correct Answer: The singular solution
Explanation:Geometrically, the singular solution of Clairaut's equation is the envelope of the family of lines defined by the general solution.
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38Solve for : .
A.
B.
C.
D.
Correct Answer:
Explanation:Use the quadratic formula for : .
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39For the equation , what is ?
A.
B.
C.
D.$0$
Correct Answer:
Explanation:. Differentiating partially w.r.t treats as constant. .
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40The equation has the singular solution determined by:
A.Replacing with
B.Differentiating the general solution w.r.t and eliminating
C.Setting
D.Setting
Correct Answer: Differentiating the general solution w.r.t and eliminating
Explanation:This is the standard procedure for finding the envelope (singular solution) from the general solution.
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41Integrating factor of is:
A.
B.
C.
D.
Correct Answer:
Explanation:Let's test I.F = . multiply eq by : . . . Exact. So is the I.F.
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42Which of these is NOT an integrating factor rule?
A.If homogeneous,
B.If ,
C.If
D.If
Correct Answer: If
Explanation:The correct rule involving division by is (or depending on sign convention used for integral). The option is generally incorrect syntax for the standard rules (Usually involves mixed partials ). Note: Standard rules are for and for .
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43The substitution and is often useful for equations of the form:
A.Linear equations
B.Exact equations
C.Clairaut's equation reducible forms (like variants)
D.Homogeneous first degree
Correct Answer: Clairaut's equation reducible forms (like variants)
Explanation:This substitution is specifically used to transform certain non-linear first-order equations into the linear Clairaut form.
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44In the general solution of a differential equation of first order, how many arbitrary constants are there?
A.
B.1
C.2
D.Depend on the degree
Correct Answer: 1
Explanation:A first-order differential equation general solution contains exactly one arbitrary constant.
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45Solve .
A.
B.
C.
D.
Correct Answer:
Explanation:This is a Clairaut equation. Replace with .
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46What is the condition for to be exact?
A.It is already exact
B.It cannot be made exact
C.Needs grouping of terms
D.None of the above
Correct Answer: Needs grouping of terms
Explanation:This looks like a mix of terms. Standard exactness check on the whole expression is the rigorous way, but often these require grouping like to become exact. The raw equation is likely not exact.
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47The discriminant relation used to find Singular Solutions from the general solution is called:
A.p-discriminant
B.c-discriminant
C.x-discriminant
D.y-discriminant
Correct Answer: c-discriminant
Explanation:Since it eliminates the constant from the quadratic in , it is the c-discriminant.
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48Equations of the form (variable is absent) are solved by:
A.Solving for and integrating
B.Differentiating w.r.t
C.Put
D.None of these
Correct Answer: Solving for and integrating
Explanation:If is missing, we solve for to get , then and integrate.
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49Solve the equation solvable for : .
A.Differentiate w.r.t to get
B.Differentiate w.r.t
C.Use quadratic formula
D.Replace with
Correct Answer: Differentiate w.r.t to get
Explanation:Standard method for 'solvable for ': Differentiate w.r.t . .
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50Find for the equation .
A.
B.
C.
D.
Correct Answer:
Explanation:. .
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