Learning Materials
Features
Discover
Chapter 12: Q82SE (page 540)
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.
The differential equation is \(\frac{{dy}}{{dx}} = {x^2} + {y^2}\).
Over 30 million students worldwide already upgrade their learning with 91影视!
All the tools & learning materials you need for study success - in one app.
Get started for free
In Problems 21鈥�24 verify that the indicated family of functions is a solution of the given differential equation. Assume an appropriate interval I of definition for each solution.
\(\frac{{dP}}{{dt}} = P(1 - P);\;P = \frac{{{c_1}{e^t}}}{{1 + {c_1}{e^t}}}\)
(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.
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 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\]
In Problems 27鈥�30 use (12) of Section 1.1 to verify that the indicated function is a solution of the given differential equation. Assume an appropriate interval I of definition of each solution.
\({x^2}y'' + \left( {{x^2} - x} \right)y' + (1 - x)y = 0;\;\;\;y = x\int_1^x {\frac{{{e^{ - t}}}}{t}} dt\)
What do you think about this solution?
We value your feedback to improve our textbook solutions.
Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine