Exercise 9.1 answers: Differential Equations

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1. The derivatives in the equation are d⁴y/dx⁴ (the fourth derivative) and y′′′ (the third derivative).
2. The highest of these is d⁴y/dx⁴, so the order is 4.
3. The third derivative y′′′ sits inside sin( ). A derivative inside a sine cannot be reached by adding, multiplying or whole-number powers, so the equation is not a polynomial in its derivatives.
4. So the degree is not defined.

AnswerOrder = 4, degree not defined.

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1. The only derivative in the equation is y′, the first derivative, so the order is 1.
2. The equation is a polynomial in y′, and y′ appears to the power 1, so the degree is 1.

AnswerOrder = 1, degree = 1.

Watch this explained “Three worked instances”, 1:32 into Order and degree, and the polynomial condition degree needs before it means anything

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1. The unknown here is s, a function of t. Its derivatives in the equation are ds/dt (first) and d²s/dt² (second).
2. The highest is d²s/dt², so the order is 2.
3. The equation uses only whole-number powers and products of its derivatives, so it is a polynomial in them.
4. The power 4 sits on ds/dt, which is not the highest derivative. The highest derivative d²s/dt² appears to the power 1, so the degree is 1.

AnswerOrder = 2, degree = 1.

Watch this explained “What degree measures”, 6:11 into Order and degree, and the polynomial condition degree needs before it means anything

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1. The derivatives in the equation are d²y/dx² and dy/dx. The highest is d²y/dx², so the order is 2.
2. The first derivative dy/dx sits inside cos( ), so the equation is not a polynomial in its derivatives.
3. So the degree is not defined, even though d²y/dx² is squared.

AnswerOrder = 2, degree not defined.

Watch this explained “Three test equations”, 5:14 into Order and degree, and the polynomial condition degree needs before it means anything

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1. The only derivative in the equation is d²y/dx², so the order is 2.
2. cos 3x and sin 3x contain only x, not a derivative, so the equation is still a polynomial in d²y/dx².
3. d²y/dx² appears to the power 1, so the degree is 1.

AnswerOrder = 2, degree = 1.

Watch this explained “What degree measures”, 6:11 into Order and degree, and the polynomial condition degree needs before it means anything

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1. The derivatives in the equation are y′′′ (third), y′′ (second) and y′ (first).
2. The highest is y′′′, so the order is 3.
3. Every derivative appears only with whole-number powers, so the equation is a polynomial in its derivatives.
4. The highest derivative y′′′ carries the power 2, so the degree is 2. The larger powers 3, 4 and 5 sit on other terms and do not decide the degree.

AnswerOrder = 3, degree = 2.

Watch this explained “What degree measures”, 6:11 into Order and degree, and the polynomial condition degree needs before it means anything

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1. The derivatives in the equation are y′′′ (third), y′′ (second) and y′ (first).
2. The highest is y′′′, so the order is 3.
3. The equation is a polynomial in its derivatives, and y′′′ appears to the power 1, so the degree is 1.

AnswerOrder = 3, degree = 1.

Watch this explained “Three worked instances”, 1:32 into Order and degree, and the polynomial condition degree needs before it means anything

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1. The only derivative in the equation is y′, so the order is 1.
2. eˣ contains no derivative, so the equation is a polynomial in y′. y′ appears to the power 1, so the degree is 1.

AnswerOrder = 1, degree = 1.

Watch this explained “Three worked instances”, 1:32 into Order and degree, and the polynomial condition degree needs before it means anything

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1. The derivatives in the equation are y′′ and y′. The highest is y′′, so the order is 2.
2. The equation is a polynomial in its derivatives.
3. (y′)² has the power 2, but y′ is not the highest derivative. y′′ appears to the power 1, so the degree is 1.

AnswerOrder = 2, degree = 1.

Watch this explained “What degree measures”, 6:11 into Order and degree, and the polynomial condition degree needs before it means anything

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1. The derivatives in the equation are y′′ and y′. The highest is y′′, so the order is 2.
2. sin y wraps y itself, not a derivative of y, so the equation is still a polynomial in its derivatives.
3. y′′ appears to the power 1, so the degree is 1.

AnswerOrder = 2, degree = 1.

Watch this explained “Three test equations”, 5:14 into Order and degree, and the polynomial condition degree needs before it means anything

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1. The order is fixed by d²y/dx², so check whether the whole equation is a polynomial in its derivatives.
2. The term sin(dy/dx) has a derivative sitting inside a sine, so the equation is not a polynomial in its derivatives — the degree does not exist.

Answer(D) not defined

Watch this explained “Three test equations”, 5:14 into Order and degree, and the polynomial condition degree needs before it means anything

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1. Find the highest derivative of y present in the equation.
2. d²y/dx² is the highest one used, so the order is 2.

Answer(A) 2

Watch this explained “Order is a count”, 0:44 into Order and degree, and the polynomial condition degree needs before it means anything