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Chapter 12: Q26E (page 509)

In Problems 25鈥�28 use (12) to verify that the indicated function is a solution of the given differential equation. Assume an appropriate interval I of definition of each solution.

\(2x\frac{{dy}}{{dx}} - y = 2xcosx;y = \sqrt x \int_4^x {\frac{{cost}}{{\sqrt t }}} dt\]

Short Answer

Expert verified

The indicated function is a solution of the differential function.

Step by step solution

01

Simplify the given differential equation.

Let the given differential equation be\(y = \sqrt x \int_4^x {\frac{{cost}}{{\sqrt t }}} dt\].

Multiply each side of the equation by\({x^{ - \frac{1}{2}}}\].

\(\begin{aligned}{c}y{x^{ - \frac{1}{2}}} = {x^{ - \frac{1}{2}}}{x^{\frac{1}{2}}}\int_4^x {\frac{{cost}}{{\sqrt t }}} \;dt\\y{x^{ - \frac{1}{2}}} = \int_4^x {\frac{{cost}}{{\sqrt t }}} \;dt\end{aligned}\]

02

Determine the solution of the indicated function.

Take differential on both sides of the equation.

Multiply\(2{x^{\frac{3}{2}}}\]on both sides of the equation.

\(\begin{aligned}{c}2x\frac{{dy}}{{\;dx}} - y = 2{x^{\frac{3}{2}}}{x^{ - \frac{1}{2}}}cosx\\2x\frac{{dy}}{{\;dx}} - y = 2xcosx\end{aligned}\]

Hence, the indicated function is a solution of the differential function.

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Most popular questions from this chapter

A tank in the form of a right-circular cylinder of radius \(2\) feet and height \(10\) feet is standing on end. If the tank is initially full of water and water leaks from a circular hole of radius \(12\) inch at its bottom, determine a differential equation for the height h of the water at time \(t > 0\). Ignore friction and contraction of water at the hole.

In Problems \(15\) and \(16\) interpret each statement as a differential equation.

On the graph of \(y = \phi (x)\) the slope of the tangent line at a point \(P(x,y)\) is the square of the distance from \(P(x,y)\) to the origin.

In Problems \(15 - 18\) verify that the indicated functionis 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.

\(y' = 25 + {y^2};y = 5tan5x\)

In Problems \(19\) and \(20\) verify that the indicated expression is an implicit solution of the given first-order differential equation. Find atleast one explicit solution \(y = \phi (x)\) in each case. Use a graphing utility to obtain the graph of an explicit solution. Give an interval \(I\) of definition of each solution \(\phi \).

(a) Verify that the one-parameter family \({y^2} - 2y = {x^2} - x + c\) is an implicit solution of the differential equation \((2y - 2)y' = 2x - 1\).

(b) Find a member of the one-parameter family in part (a) that satisfies the initial condition \(y(0) = 1\).

(c) Use your result in part (b) to and an explicit function \(y = \phi (x)\) that satisfies \(y(0) = 1\). Give the domain of the function \(\phi \). Is \(y = \phi (x)\) a solution of the initial-value problem? If so, give its interval \(I\) of definition; if not, explain.
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