Why canâÂÂt we apply Method of separation of variables to second order Diff equations?
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Suppose we are solving the following ODE $$mathrmdyover mathrmdx=yover x$$ then we can solve it by seperation of variables method. But if we have to solve $$mathrmd^2yover mathrmdx^2 = yover x$$ then why cant we seperate $mathrmd^2yover mathrmdx^2$?
differential-equations
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Suppose we are solving the following ODE $$mathrmdyover mathrmdx=yover x$$ then we can solve it by seperation of variables method. But if we have to solve $$mathrmd^2yover mathrmdx^2 = yover x$$ then why cant we seperate $mathrmd^2yover mathrmdx^2$?
differential-equations
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
â José Carlos Santos
Aug 20 at 11:32
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up vote
5
down vote
favorite
up vote
5
down vote
favorite
Suppose we are solving the following ODE $$mathrmdyover mathrmdx=yover x$$ then we can solve it by seperation of variables method. But if we have to solve $$mathrmd^2yover mathrmdx^2 = yover x$$ then why cant we seperate $mathrmd^2yover mathrmdx^2$?
differential-equations
Suppose we are solving the following ODE $$mathrmdyover mathrmdx=yover x$$ then we can solve it by seperation of variables method. But if we have to solve $$mathrmd^2yover mathrmdx^2 = yover x$$ then why cant we seperate $mathrmd^2yover mathrmdx^2$?
differential-equations
edited Aug 20 at 11:44
Bernard
111k635103
111k635103
asked Aug 20 at 11:29
megha gill
261
261
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
â José Carlos Santos
Aug 20 at 11:32
add a comment |Â
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
â José Carlos Santos
Aug 20 at 11:32
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
â José Carlos Santos
Aug 20 at 11:32
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
â José Carlos Santos
Aug 20 at 11:32
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2 Answers
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You have to be cautious: remember that $$fracd^2ydx^2 = fracdleft(fracdydxright)dx$$ So you could use separation of variables, but your ODE will become $$frac1ydleft(fracdydxright) = frac1xdx$$ and not $$frac1ydydy = frac1xdxdx;;;;colorredtextWrong$$
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Because you can't write $$dfracd^2 yy=dfracdx^2x$$or equivalently $$dfracd^2yy=dfracdx cdot dxx$$this is meaningless. Also you would be stopped in the next step. $int dfracd^2yy$ and $intdfracdx^2x$ are both illegal.
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
You have to be cautious: remember that $$fracd^2ydx^2 = fracdleft(fracdydxright)dx$$ So you could use separation of variables, but your ODE will become $$frac1ydleft(fracdydxright) = frac1xdx$$ and not $$frac1ydydy = frac1xdxdx;;;;colorredtextWrong$$
add a comment |Â
up vote
4
down vote
You have to be cautious: remember that $$fracd^2ydx^2 = fracdleft(fracdydxright)dx$$ So you could use separation of variables, but your ODE will become $$frac1ydleft(fracdydxright) = frac1xdx$$ and not $$frac1ydydy = frac1xdxdx;;;;colorredtextWrong$$
add a comment |Â
up vote
4
down vote
up vote
4
down vote
You have to be cautious: remember that $$fracd^2ydx^2 = fracdleft(fracdydxright)dx$$ So you could use separation of variables, but your ODE will become $$frac1ydleft(fracdydxright) = frac1xdx$$ and not $$frac1ydydy = frac1xdxdx;;;;colorredtextWrong$$
You have to be cautious: remember that $$fracd^2ydx^2 = fracdleft(fracdydxright)dx$$ So you could use separation of variables, but your ODE will become $$frac1ydleft(fracdydxright) = frac1xdx$$ and not $$frac1ydydy = frac1xdxdx;;;;colorredtextWrong$$
answered Aug 20 at 11:42
Davide Morgante
2,343422
2,343422
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add a comment |Â
up vote
0
down vote
Because you can't write $$dfracd^2 yy=dfracdx^2x$$or equivalently $$dfracd^2yy=dfracdx cdot dxx$$this is meaningless. Also you would be stopped in the next step. $int dfracd^2yy$ and $intdfracdx^2x$ are both illegal.
add a comment |Â
up vote
0
down vote
Because you can't write $$dfracd^2 yy=dfracdx^2x$$or equivalently $$dfracd^2yy=dfracdx cdot dxx$$this is meaningless. Also you would be stopped in the next step. $int dfracd^2yy$ and $intdfracdx^2x$ are both illegal.
add a comment |Â
up vote
0
down vote
up vote
0
down vote
Because you can't write $$dfracd^2 yy=dfracdx^2x$$or equivalently $$dfracd^2yy=dfracdx cdot dxx$$this is meaningless. Also you would be stopped in the next step. $int dfracd^2yy$ and $intdfracdx^2x$ are both illegal.
Because you can't write $$dfracd^2 yy=dfracdx^2x$$or equivalently $$dfracd^2yy=dfracdx cdot dxx$$this is meaningless. Also you would be stopped in the next step. $int dfracd^2yy$ and $intdfracdx^2x$ are both illegal.
answered Aug 20 at 11:44
Mostafa Ayaz
9,7483730
9,7483730
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Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
â José Carlos Santos
Aug 20 at 11:32