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Chapter 12: Q57E (page 534)

In Problems 7鈥�12 match each of the given differential equations with one or more of these solutions:

(a) \(y = 0\), (b) \(y = 2\), (c) \(y = 2x\), (d) \(y = 2{x^2}\)

\(y' = 2y - 4\)

Short Answer

Expert verified

The solution is \({\rm{y = 2}}\).

Step by step solution

01

Check \(y = 0\) if is a solution.

Put\({\rm{y = 0}}\).

\(\begin{array}{c}(0)' = 2(0) - 4\\0 = - 4\\0 \ne - 4\end{array}\)

Hence, it is not a solution.

02

Check \(y = 2\) if is a solution.

Put\({\rm{y = 2}}\).

\(\begin{array}{c}(2)' = 2(2) - 4\\0 = 4 - 4\\0 = 0\end{array}\)

Hence, it is a solution.

03

Check \(y = 2x\) if is a solution.

Put\({\rm{y = 2x}}\).

\(\begin{array}{c}(2x)' = 2(2x) - 4\\2 = 4x - 4\\2 \ne 4x - 4\end{array}\)

Hence, it is a solution.

04

Check \(y = 2{x^2}\) if is a solution.

Put \({\rm{y = 2}}{{\rm{x}}^2}\).

\(\begin{array}{c}(2{x^2})' = 2(2{x^2}) - 4\\4x = 4{x^2} - 4\\4x \ne 4{x^2} - 4\end{array}\)

Hence, it is not a solution.

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Most popular questions from this chapter

In Problems \(15 - 18\) verify that the indicated function \(y = \phi (x)\) is an explicit solution of the given first-order differential equation. Proceed as in Example \(6\), by considering \(\phi \) simply as a function and give its domain. Then by considering \(\phi \) as a solution of the differential equation, give at least one interval \(I\) of definition.

\(y' = 2x{y^2};y = 1/(4 - {x^2})\)

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