How can I know how many solutions of a differential equation passing a point?
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I was solving this problem from Mathematical Methods, by Mary L. Boas, but I couldn't prove that there is only one solution through any point for which $y>0$. I go back to the text and read it many times, but I still can prove it.
By separation of variables, find a solution of the equation $y^{prime} = sqrt{y}$ containing one arbitrary constant. Find a particular solution satisfying $y = 0$ when $x = 0$. Show that $y = 0$ is a solution of the differential equation which cannot be obtained by specializing the arbitrary constant in your solution above. Computer plot a slope field and some of the solution curves. Show that there is an infinite number of solution curves passing through any point on the x-axis, but just one through any point for which $y > 0$. Hint: See Example 3. Problems 17 and 18 are physical problems leading to this differential equation.
I know that the slope is positive, but I couldn't use it since I keep getting different values of the constant for any value of $y>0$.
differential-equations
edited Nov 26 '18 at 18:25
Did
246k23221456
asked Nov 24 '18 at 2:02
David ScottDavid Scott
82
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add a comment |
$begingroup$
I was solving this problem from Mathematical Methods, by Mary L. Boas, but I couldn't prove that there is only one solution through any point for which $y>0$. I go back to the text and read it many times, but I still can prove it.
By separation of variables, find a solution of the equation $y^{prime} = sqrt{y}$ containing one arbitrary constant. Find a particular solution satisfying $y = 0$ when $x = 0$. Show that $y = 0$ is a solution of the differential equation which cannot be obtained by specializing the arbitrary constant in your solution above. Computer plot a slope field and some of the solution curves. Show that there is an infinite number of solution curves passing through any point on the x-axis, but just one through any point for which $y > 0$. Hint: See Example 3. Problems 17 and 18 are physical problems leading to this differential equation.
I know that the slope is positive, but I couldn't use it since I keep getting different values of the constant for any value of $y>0$.
differential-equations
edited Nov 26 '18 at 18:25
Did
246k23221456
asked Nov 24 '18 at 2:02
David ScottDavid Scott
82
$endgroup$
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Hint: Cauchy-Lipschitz.
$endgroup$
– Did
Nov 26 '18 at 18:25
add a comment |
$begingroup$
I was solving this problem from Mathematical Methods, by Mary L. Boas, but I couldn't prove that there is only one solution through any point for which $y>0$. I go back to the text and read it many times, but I still can prove it.
By separation of variables, find a solution of the equation $y^{prime} = sqrt{y}$ containing one arbitrary constant. Find a particular solution satisfying $y = 0$ when $x = 0$. Show that $y = 0$ is a solution of the differential equation which cannot be obtained by specializing the arbitrary constant in your solution above. Computer plot a slope field and some of the solution curves. Show that there is an infinite number of solution curves passing through any point on the x-axis, but just one through any point for which $y > 0$. Hint: See Example 3. Problems 17 and 18 are physical problems leading to this differential equation.
I know that the slope is positive, but I couldn't use it since I keep getting different values of the constant for any value of $y>0$.
differential-equations
edited Nov 26 '18 at 18:25
Did
246k23221456
asked Nov 24 '18 at 2:02
David ScottDavid Scott
82
$endgroup$
I was solving this problem from Mathematical Methods, by Mary L. Boas, but I couldn't prove that there is only one solution through any point for which $y>0$. I go back to the text and read it many times, but I still can prove it.
By separation of variables, find a solution of the equation $y^{prime} = sqrt{y}$ containing one arbitrary constant. Find a particular solution satisfying $y = 0$ when $x = 0$. Show that $y = 0$ is a solution of the differential equation which cannot be obtained by specializing the arbitrary constant in your solution above. Computer plot a slope field and some of the solution curves. Show that there is an infinite number of solution curves passing through any point on the x-axis, but just one through any point for which $y > 0$. Hint: See Example 3. Problems 17 and 18 are physical problems leading to this differential equation.
I know that the slope is positive, but I couldn't use it since I keep getting different values of the constant for any value of $y>0$.
differential-equations
differential-equations
edited Nov 26 '18 at 18:25
Did
246k23221456
asked Nov 24 '18 at 2:02
David ScottDavid Scott
82
edited Nov 26 '18 at 18:25
Did
246k23221456
asked Nov 24 '18 at 2:02
David ScottDavid Scott
82
edited Nov 26 '18 at 18:25
Did
246k23221456
edited Nov 26 '18 at 18:25
Did
246k23221456
edited Nov 26 '18 at 18:25
Did
246k23221456
246k23221456
asked Nov 24 '18 at 2:02
David ScottDavid Scott
82
asked Nov 24 '18 at 2:02
David ScottDavid Scott
82
asked Nov 24 '18 at 2:02
David ScottDavid Scott
82
82
$begingroup$
Hint: Cauchy-Lipschitz.
$endgroup$
– Did
Nov 26 '18 at 18:25
add a comment |
$begingroup$
Hint: Cauchy-Lipschitz.
$endgroup$
– Did
Nov 26 '18 at 18:25
$begingroup$
Hint: Cauchy-Lipschitz.
$endgroup$
– Did
Nov 26 '18 at 18:25
$begingroup$
Hint: Cauchy-Lipschitz.
$endgroup$
– Did
Nov 26 '18 at 18:25
add a comment |
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$begingroup$
Hint: Cauchy-Lipschitz.
$endgroup$
– Did
Nov 26 '18 at 18:25