Discrepancy in solutions of differential equation?
up vote
1
down vote
favorite
The differential equation at hand is this :
$$ frac{text{d}psi}{text{d}x}+2tanh(x),psileft(xright)=0 $$
And what I have tried is this :
$$
int_{}^{} frac{text{d}psi}{psi}=-2int_{}^{} tanh(x),dx$$
and
$$ lnpsi left(xright)=-2cosh^{-2}left(xright)+C $$
And the solution of this elementary problem comes out to be :
$$ psileft(xright)=Ae^{-2cosh^{-2}left(xright)}$$
But clearly,
$$ psileft(xright) = cosh^{-2}left(xright) $$
is a solution. But why can't I find it through integration?
differential-equations hyperbolic-functions
edited Nov 14 at 11:19
Robert Z
90k1056128
asked Nov 14 at 11:00
Jalaj Chaturvedi
787
add a comment |
up vote
1
down vote
favorite
The differential equation at hand is this :
$$ frac{text{d}psi}{text{d}x}+2tanh(x),psileft(xright)=0 $$
And what I have tried is this :
$$
int_{}^{} frac{text{d}psi}{psi}=-2int_{}^{} tanh(x),dx$$
and
$$ lnpsi left(xright)=-2cosh^{-2}left(xright)+C $$
And the solution of this elementary problem comes out to be :
$$ psileft(xright)=Ae^{-2cosh^{-2}left(xright)}$$
But clearly,
$$ psileft(xright) = cosh^{-2}left(xright) $$
is a solution. But why can't I find it through integration?
differential-equations hyperbolic-functions
edited Nov 14 at 11:19
Robert Z
90k1056128
asked Nov 14 at 11:00
Jalaj Chaturvedi
787
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
The differential equation at hand is this :
$$ frac{text{d}psi}{text{d}x}+2tanh(x),psileft(xright)=0 $$
And what I have tried is this :
$$
int_{}^{} frac{text{d}psi}{psi}=-2int_{}^{} tanh(x),dx$$
and
$$ lnpsi left(xright)=-2cosh^{-2}left(xright)+C $$
And the solution of this elementary problem comes out to be :
$$ psileft(xright)=Ae^{-2cosh^{-2}left(xright)}$$
But clearly,
$$ psileft(xright) = cosh^{-2}left(xright) $$
is a solution. But why can't I find it through integration?
differential-equations hyperbolic-functions
edited Nov 14 at 11:19
Robert Z
90k1056128
asked Nov 14 at 11:00
Jalaj Chaturvedi
787
The differential equation at hand is this :
$$ frac{text{d}psi}{text{d}x}+2tanh(x),psileft(xright)=0 $$
And what I have tried is this :
$$
int_{}^{} frac{text{d}psi}{psi}=-2int_{}^{} tanh(x),dx$$
and
$$ lnpsi left(xright)=-2cosh^{-2}left(xright)+C $$
And the solution of this elementary problem comes out to be :
$$ psileft(xright)=Ae^{-2cosh^{-2}left(xright)}$$
But clearly,
$$ psileft(xright) = cosh^{-2}left(xright) $$
is a solution. But why can't I find it through integration?
differential-equations hyperbolic-functions
differential-equations hyperbolic-functions
edited Nov 14 at 11:19
Robert Z
90k1056128
asked Nov 14 at 11:00
Jalaj Chaturvedi
787
edited Nov 14 at 11:19
Robert Z
90k1056128
asked Nov 14 at 11:00
Jalaj Chaturvedi
787
edited Nov 14 at 11:19
Robert Z
90k1056128
edited Nov 14 at 11:19
Robert Z
90k1056128
edited Nov 14 at 11:19
Robert Z
90k1056128
90k1056128
asked Nov 14 at 11:00
Jalaj Chaturvedi
787
asked Nov 14 at 11:00
Jalaj Chaturvedi
787
asked Nov 14 at 11:00
Jalaj Chaturvedi
787
787
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
up vote
1
down vote
accepted
Actually, we have that
$$inttanh(x) dx=intfrac{d(cosh(x))}{cosh(x)}=ln(cosh(x))+c$$
and
$$D(tanh(x))=frac{1}{cosh^2(x)}.$$
So you confused the derivative with the integral...
answered Nov 14 at 11:04
Robert Z
90k1056128
Oh! Thanks, I didn't thought about that.
– Jalaj Chaturvedi
Nov 14 at 11:12
@JalajChaturvedi Well done. BTW, if you are new here, please take a few minutes for a tour: math.stackexchange.com/tour
– Robert Z
Nov 14 at 11:32
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
Actually, we have that
$$inttanh(x) dx=intfrac{d(cosh(x))}{cosh(x)}=ln(cosh(x))+c$$
and
$$D(tanh(x))=frac{1}{cosh^2(x)}.$$
So you confused the derivative with the integral...
answered Nov 14 at 11:04
Robert Z
90k1056128
Oh! Thanks, I didn't thought about that.
– Jalaj Chaturvedi
Nov 14 at 11:12
@JalajChaturvedi Well done. BTW, if you are new here, please take a few minutes for a tour: math.stackexchange.com/tour
– Robert Z
Nov 14 at 11:32
add a comment |
up vote
1
down vote
accepted
Actually, we have that
$$inttanh(x) dx=intfrac{d(cosh(x))}{cosh(x)}=ln(cosh(x))+c$$
and
$$D(tanh(x))=frac{1}{cosh^2(x)}.$$
So you confused the derivative with the integral...
answered Nov 14 at 11:04
Robert Z
90k1056128
Oh! Thanks, I didn't thought about that.
– Jalaj Chaturvedi
Nov 14 at 11:12
@JalajChaturvedi Well done. BTW, if you are new here, please take a few minutes for a tour: math.stackexchange.com/tour
– Robert Z
Nov 14 at 11:32
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
Actually, we have that
$$inttanh(x) dx=intfrac{d(cosh(x))}{cosh(x)}=ln(cosh(x))+c$$
and
$$D(tanh(x))=frac{1}{cosh^2(x)}.$$
So you confused the derivative with the integral...
answered Nov 14 at 11:04
Robert Z
90k1056128
Actually, we have that
$$inttanh(x) dx=intfrac{d(cosh(x))}{cosh(x)}=ln(cosh(x))+c$$
and
$$D(tanh(x))=frac{1}{cosh^2(x)}.$$
So you confused the derivative with the integral...
answered Nov 14 at 11:04
Robert Z
90k1056128
edited Nov 14 at 11:17
answered Nov 14 at 11:04
Robert Z
90k1056128
answered Nov 14 at 11:04
Robert Z
90k1056128
answered Nov 14 at 11:04
Robert Z
90k1056128
90k1056128
Oh! Thanks, I didn't thought about that.
– Jalaj Chaturvedi
Nov 14 at 11:12
@JalajChaturvedi Well done. BTW, if you are new here, please take a few minutes for a tour: math.stackexchange.com/tour
– Robert Z
Nov 14 at 11:32
add a comment |
Oh! Thanks, I didn't thought about that.
– Jalaj Chaturvedi
Nov 14 at 11:12
@JalajChaturvedi Well done. BTW, if you are new here, please take a few minutes for a tour: math.stackexchange.com/tour
– Robert Z
Nov 14 at 11:32
Oh! Thanks, I didn't thought about that.
– Jalaj Chaturvedi
Nov 14 at 11:12
Oh! Thanks, I didn't thought about that.
– Jalaj Chaturvedi
Nov 14 at 11:12
@JalajChaturvedi Well done. BTW, if you are new here, please take a few minutes for a tour: math.stackexchange.com/tour
– Robert Z
Nov 14 at 11:32
@JalajChaturvedi Well done. BTW, if you are new here, please take a few minutes for a tour: math.stackexchange.com/tour
– Robert Z
Nov 14 at 11:32
add a comment |
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2998127%2fdiscrepancy-in-solutions-of-differential-equation%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
