I am interested in the following differential equation: y″−qy where q:R+→R+⋅ is a continuous, positive function.

Expert Answer to "I am interested in the following differential equation:..."

High School
Calculus and Analysis
Calculus 2
Differential equations

Jazmyn Holden

Answered question

2022-04-01

I am interested in the following differential equation:

y^{″} - q y

where

q : R_{+} \to R_{+}^{\cdot}

is a continuous, positive function.

Answer & Explanation

memantangti17

Beginner2022-04-02Added 13 answers

Step 1

y^{″} = p y

y^{″} y^{'} = p y y^{'}

y^{' 2} = p y^{2} - \int p^{'} y^{2} d x

by integration by parts.
For a vanishingly small region about a zero of y, that integral is zero. This implies a zero of y happens for a zero of the first derivative.
The second derivative is always non-negative, so the first derivative is monotonically increasing. Once positive, the first derivative will remain positive, so it only has a zero if it has a negative initial value.
The first derivative will exceed any initial value, so any

a < 0

will work.
So

k = 0

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Calculus 2

What is the derivative of the work function?
How to use implicit differentiation to find $\frac{d y}{d x}$ given $3 x^{2} + 3 y^{2} = 2$ ?
How to differentiate $y = \log x^{2}$ ?
The solution of a differential equation y′′+3y′+2y=0 is of the form
A) $c_{1} e^{x} + c_{2} e^{2 x}$
B) $c_{1} e^{- x} + c_{2} e^{3 x}$
C) $c_{1} e^{- x} + c_{2} e^{- 2 x}$
D) $c_{1} e^{- 2 x} + c_{2} 2^{- x}$
How to find instantaneous velocity from a position vs. time graph?
How to implicitly differentiate $\sqrt{x y} = x - 2 y$ ?
What is 2xy differentiated implicitly?
How to find the sum of the infinite geometric series given $1 + \frac{2}{3} + \frac{4}{9} + ...$ ?
Look at this series: 1.5, 2.3, 3.1, 3.9, ... What number should come next?
A. 4.2
B. 4.4
C. 4.7
D. 5.1
What is the derivative of $\frac{x + 1}{y}$ ?
How to find the sum of the infinite geometric series 0.9 + 0.09 + 0.009 +…?
How to find the volume of a cone using an integral?
What is the surface area of the solid created by revolving $f (x) = e^{2 - x}, x \in [1, 2]$ around the x axis?
How to differentiate $x^{\frac{2}{3}} + y^{\frac{2}{3}} = 4$ ?
The differential coefficient of $\sec (\tan^{- 1} (x))$ .

Didn't find what you were looking for?