What an ODE is practice problems

An ODE links a function to its derivatives, for example y' = 2y. Here you classify equations by order and linearity, verify a proposed solution by substitution, solve y' = f(x) by integrating, use an initial condition to fix C, and read slope fields. Recognition cue: the problem shows y' or y'' or names a rate of change, and asks you to classify, verify, solve, or describe behavior. You will practice: Order and Linearity, Verify by Substitution, Direct Integration, Slope Field Reading.

11 problems, each with a worked answer. Work them on paper, in order — the difficulty ramps gently.

State the order of the differential equation y'' + 4y = 0.

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Answer: $2$

The highest derivative present is y''. The order is the highest derivative.

order = 2

Find the general solution of y' = 4.

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Answer: $y = 4 x + C$

The right side is a function of x alone. Integrate once and keep the constant.

y = \int 4 d x = 4 x + C

Check: y' = 4.

A slope field is drawn for y' = x + y. What is the slope of the short segment at the point (1, 2)?

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Answer: $3$

The slope at a point is the value of the right side at that point.

1 + 2 = 3

Is the differential equation y' + y^2 = x linear or nonlinear?

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Answer: $nonlinear$

The term y^2 has the unknown to the second power. Linear equations allow y and its derivatives only to the first power, with coefficients in x.

y^{'} + y^{2} = x

Find the value of k so that y = e^(kx) is a solution of y' = 5y.

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Answer: $k = 5$

Differentiate and substitute.

y^{'} = k e^{k x} = 5 e^{k x}

The factor e^(kx) is never 0, so divide it out.

k = 5

Find the general solution of y' = 6x^2 - 4.

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Answer: $y = 2 x^{3} - 4 x + C$

Integrate the right side and keep C.

y = \int (6 x^{2} - 4) d x = 2 x^{3} - 4 x + C

Check: y' = 6x^2 - 4.

A particle moves along a line with velocity v(t) = 3t^2 meters per second. Its position satisfies s' = v with s(0) = 2. Find s(t).

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Answer: $s = t^{3} + 2$

Integrate the velocity, then use s(0) = 2.

s = t^{3} + C

2 = 0 + C

s = t^{3} + 2

Find all constant solutions of y' = y(3 - y).

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Answer: $y = 0, y = 3$

A constant solution has y' = 0 for all x. Set the right side to 0.

y (3 - y) = 0

y = 0, y = 3

The general solution of y' = 2x is y = x^2 + C. Find the value of C so that y(2) = 7.

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Answer: $C = 3$

Substitute x = 2 and y = 7 into the general solution.

7 = 4 + C

C = 3

10.

A bacteria population grows at the rate P'(t) = 200t cells per hour, with P(0) = 500. Find P(t).

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Answer: $P = 100 t^{2} + 500$

Integrate, then apply P(0) = 500.

P = 100 t^{2} + C

500 = 0 + C

P = 100 t^{2} + 500

Check: P' = 200t.

11.

Consider the single equation y''' + t²y' = cos t. State its order. Then state whether it is linear or nonlinear.

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Answer: $order 3, linear$

The highest derivative present is y''', so the order is 3. Every term is y or a derivative of y to the first power, with coefficients that depend only on t. The forcing cos t involves no y. So the equation is linear.

y^{'''} + t^{2} y^{'} = cos t

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What an ODE is practice problems

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