Discuss the difference between the differential equations(dy/dx)2 = 4y and dy/dx = 2/ỹ. Do they have thesame solution curves? Why or why not? Determine thepoints (a, b) in the plane for which the initial value prob-lem y' = 2/5, y(a) = b has(b) a uniquesolution,

given two differential equation dydx2=4y and dydx=2y claim- discuss the difference between the…

Answered: Discuss the difference between the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Question

Discuss the difference between the differential equations
(dy/dx)2 = 4y and dy/dx = 2/ỹ. Do they have the
same solution curves? Why or why not? Determine the
points (a, b) in the plane for which the initial value prob-
lem y' = 2/5, y(a) = b has
(b) a unique
solution,

Expert Solution

Step 1

given two differential equation

${(\frac{d y}{d x})}^{2} = 4 y$ and $\frac{d y}{d x} = 2 \sqrt{y}$

claim- discuss the difference between the equation. Do they have the same solution curves ? why or why not ?

${(\frac{d y}{d x})}^{2} = 4 y \Rightarrow {(y')}^{2} = 4 y$

differentiate it

$2 y' y'' = 4 y' \Rightarrow y'' = 2$

integrate the function

$y' = 2 x + c_{1}$

again integrate the function

$y = x^{2} + c_{1} x + c_{2}$

Step 2

now, integrate $\frac{d y}{d x} = 2 \sqrt{y}$

using separation of variable

$\int \frac{d y}{\sqrt{y}} = \int 2 d x \Rightarrow 2 y^{\frac{1}{2}} = 2 x + c \Rightarrow y^{\frac{1}{2}} = x + \frac{c}{2}$

squaring both sides

$y = {(x + \frac{c}{2})}^{2} = x^{2} + {(\frac{c}{2})}^{2} + 2 x \times \frac{c}{2} = x^{2} + x c + \frac{c^{2}}{4}$

let $\frac{c^{2}}{4} = c_{2}, c = c_{1}$

$y = x^{2} + x c_{1} + c_{2}$

on solving the two equation gives same curve

that is , both the equations

${(\frac{d y}{d x})}^{2} = 4 y$ and $\frac{d y}{d x} = 2 \sqrt{y}$

have same solution.

Step by step

Solved in 3 steps

Similar questions

Recommended textbooks for you

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated

Numerical Methods for Engineers

Advanced Math

ISBN:

9780073397924

Author:

Steven C. Chapra Dr., Raymond P. Canale

Publisher:

McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…

Advanced Math

ISBN:

9781118141809

Author:

Nathan Klingbeil

Publisher:

WILEY

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated

Numerical Methods for Engineers

Advanced Math

ISBN:

9780073397924

Author:

Steven C. Chapra Dr., Raymond P. Canale

Publisher:

McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…

Advanced Math

ISBN:

9781118141809

Author:

Nathan Klingbeil

Publisher:

WILEY

Mathematics For Machine Technology

Advanced Math

ISBN:

9781337798310

Author:

Peterson, John.

Publisher:

Cengage Learning,

Basic Technical Mathematics

Advanced Math

ISBN:

9780134437705

Author:

Washington

Publisher:

PEARSON

Topology

Advanced Math

ISBN:

9780134689517

Author:

Munkres, James R.

Publisher:

Pearson,

SEE MORE TEXTBOOKS

Discuss the difference between the differential equations (dy/dx)2 = 4y and dy/dx = 2/ỹ. Do they have the same solution curves? Why or why not? Determine the points (a, b) in the plane for which the initial value prob- lem y' = 2/5, y(a) = b has (b) a unique solution,