Determine a solution of an ODE by “inspection”
I'm checking Zill's A First Course in Differential Equations with Modeling Applications, and there's an exercise that says:
From the following problems determine by inspection at least two solutions of the given IVP.
- $y'=3y^{2/3},,y(0)=0$
- $xy'=2y,,y(0)=0$
I don't quite understand what in means by "by inspection", what's the difference between just finding the solutions and determine by inspection?
differential-equations
asked Aug 12 '16 at 18:00
Ana Galois
1,2401233
add a comment |
I'm checking Zill's A First Course in Differential Equations with Modeling Applications, and there's an exercise that says:
From the following problems determine by inspection at least two solutions of the given IVP.
- $y'=3y^{2/3},,y(0)=0$
- $xy'=2y,,y(0)=0$
I don't quite understand what in means by "by inspection", what's the difference between just finding the solutions and determine by inspection?
differential-equations
asked Aug 12 '16 at 18:00
Ana Galois
1,2401233
1
The difference is doing it in your head. You can ignore that stipulation if you want.
– arctic tern
Aug 12 '16 at 18:04
add a comment |
I'm checking Zill's A First Course in Differential Equations with Modeling Applications, and there's an exercise that says:
From the following problems determine by inspection at least two solutions of the given IVP.
- $y'=3y^{2/3},,y(0)=0$
- $xy'=2y,,y(0)=0$
I don't quite understand what in means by "by inspection", what's the difference between just finding the solutions and determine by inspection?
differential-equations
asked Aug 12 '16 at 18:00
Ana Galois
1,2401233
I'm checking Zill's A First Course in Differential Equations with Modeling Applications, and there's an exercise that says:
From the following problems determine by inspection at least two solutions of the given IVP.
- $y'=3y^{2/3},,y(0)=0$
- $xy'=2y,,y(0)=0$
I don't quite understand what in means by "by inspection", what's the difference between just finding the solutions and determine by inspection?
differential-equations
differential-equations
asked Aug 12 '16 at 18:00
Ana Galois
1,2401233
asked Aug 12 '16 at 18:00
Ana Galois
1,2401233
asked Aug 12 '16 at 18:00
Ana Galois
1,2401233
asked Aug 12 '16 at 18:00
Ana Galois
1,2401233
asked Aug 12 '16 at 18:00
Ana Galois
1,2401233
1,2401233
1
The difference is doing it in your head. You can ignore that stipulation if you want.
– arctic tern
Aug 12 '16 at 18:04
add a comment |
1
The difference is doing it in your head. You can ignore that stipulation if you want.
– arctic tern
Aug 12 '16 at 18:04
1
1
The difference is doing it in your head. You can ignore that stipulation if you want.
– arctic tern
Aug 12 '16 at 18:04
The difference is doing it in your head. You can ignore that stipulation if you want.
– arctic tern
Aug 12 '16 at 18:04
add a comment |
3 Answers
3
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In general, the phrase "by inspection" means "by reasonable guessing". For instance, suppose we are to prove that there are integers $x,y,z$ such that $x^{2}+y^{2}=z^{2}$. Without manipulating the given condition, we have a candidate solution to the equation in mind, i.e. the triple $(3,4,5)$ of integers, which happens to be a solution to the equation; hence the proposition is proved. Well, we just proved the proposition by inspection.
Yes, it is just the same thing to say "just find some solutions" instead of "find some solutions by inspection".
answered Aug 12 '16 at 18:36
Gary Moore
17.2k21545
add a comment |
To find a solution ''by inspection'' usually means that the solution can be found (almost) immediately, using some result that is expected to be well known.
In your case, as an example, from the fact that the derivative of $y=x^2$ is $y'=2x$ we can say ''by inspection'' that the solution of $xy'=2y$ can be a function of the form $y=x^2$, that satisfies the initial condition.
Analogously, using the fact that $y=x^3 rightarrow y'=3x^2$ you can solve ''by inspection'' the other IVP.
answered Aug 12 '16 at 19:52
Emilio Novati
51.4k43472
add a comment |
I already see that the trivial solution $y=0$ satisfies both systems. We can slso try $y=x^3$ for the first and $y=x^2$ for the second.
answered Nov 23 at 14:23
Wesley Strik
1,486422
add a comment |
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
In general, the phrase "by inspection" means "by reasonable guessing". For instance, suppose we are to prove that there are integers $x,y,z$ such that $x^{2}+y^{2}=z^{2}$. Without manipulating the given condition, we have a candidate solution to the equation in mind, i.e. the triple $(3,4,5)$ of integers, which happens to be a solution to the equation; hence the proposition is proved. Well, we just proved the proposition by inspection.
Yes, it is just the same thing to say "just find some solutions" instead of "find some solutions by inspection".
answered Aug 12 '16 at 18:36
Gary Moore
17.2k21545
add a comment |
In general, the phrase "by inspection" means "by reasonable guessing". For instance, suppose we are to prove that there are integers $x,y,z$ such that $x^{2}+y^{2}=z^{2}$. Without manipulating the given condition, we have a candidate solution to the equation in mind, i.e. the triple $(3,4,5)$ of integers, which happens to be a solution to the equation; hence the proposition is proved. Well, we just proved the proposition by inspection.
Yes, it is just the same thing to say "just find some solutions" instead of "find some solutions by inspection".
answered Aug 12 '16 at 18:36
Gary Moore
17.2k21545
add a comment |
In general, the phrase "by inspection" means "by reasonable guessing". For instance, suppose we are to prove that there are integers $x,y,z$ such that $x^{2}+y^{2}=z^{2}$. Without manipulating the given condition, we have a candidate solution to the equation in mind, i.e. the triple $(3,4,5)$ of integers, which happens to be a solution to the equation; hence the proposition is proved. Well, we just proved the proposition by inspection.
Yes, it is just the same thing to say "just find some solutions" instead of "find some solutions by inspection".
answered Aug 12 '16 at 18:36
Gary Moore
17.2k21545
In general, the phrase "by inspection" means "by reasonable guessing". For instance, suppose we are to prove that there are integers $x,y,z$ such that $x^{2}+y^{2}=z^{2}$. Without manipulating the given condition, we have a candidate solution to the equation in mind, i.e. the triple $(3,4,5)$ of integers, which happens to be a solution to the equation; hence the proposition is proved. Well, we just proved the proposition by inspection.
Yes, it is just the same thing to say "just find some solutions" instead of "find some solutions by inspection".
answered Aug 12 '16 at 18:36
Gary Moore
17.2k21545
answered Aug 12 '16 at 18:36
Gary Moore
17.2k21545
answered Aug 12 '16 at 18:36
Gary Moore
17.2k21545
answered Aug 12 '16 at 18:36
Gary Moore
17.2k21545
17.2k21545
add a comment |
add a comment |
To find a solution ''by inspection'' usually means that the solution can be found (almost) immediately, using some result that is expected to be well known.
In your case, as an example, from the fact that the derivative of $y=x^2$ is $y'=2x$ we can say ''by inspection'' that the solution of $xy'=2y$ can be a function of the form $y=x^2$, that satisfies the initial condition.
Analogously, using the fact that $y=x^3 rightarrow y'=3x^2$ you can solve ''by inspection'' the other IVP.
answered Aug 12 '16 at 19:52
Emilio Novati
51.4k43472
add a comment |
To find a solution ''by inspection'' usually means that the solution can be found (almost) immediately, using some result that is expected to be well known.
In your case, as an example, from the fact that the derivative of $y=x^2$ is $y'=2x$ we can say ''by inspection'' that the solution of $xy'=2y$ can be a function of the form $y=x^2$, that satisfies the initial condition.
Analogously, using the fact that $y=x^3 rightarrow y'=3x^2$ you can solve ''by inspection'' the other IVP.
answered Aug 12 '16 at 19:52
Emilio Novati
51.4k43472
add a comment |
To find a solution ''by inspection'' usually means that the solution can be found (almost) immediately, using some result that is expected to be well known.
In your case, as an example, from the fact that the derivative of $y=x^2$ is $y'=2x$ we can say ''by inspection'' that the solution of $xy'=2y$ can be a function of the form $y=x^2$, that satisfies the initial condition.
Analogously, using the fact that $y=x^3 rightarrow y'=3x^2$ you can solve ''by inspection'' the other IVP.
answered Aug 12 '16 at 19:52
Emilio Novati
51.4k43472
To find a solution ''by inspection'' usually means that the solution can be found (almost) immediately, using some result that is expected to be well known.
In your case, as an example, from the fact that the derivative of $y=x^2$ is $y'=2x$ we can say ''by inspection'' that the solution of $xy'=2y$ can be a function of the form $y=x^2$, that satisfies the initial condition.
Analogously, using the fact that $y=x^3 rightarrow y'=3x^2$ you can solve ''by inspection'' the other IVP.
answered Aug 12 '16 at 19:52
Emilio Novati
51.4k43472
edited Aug 12 '16 at 20:13
answered Aug 12 '16 at 19:52
Emilio Novati
51.4k43472
answered Aug 12 '16 at 19:52
Emilio Novati
51.4k43472
answered Aug 12 '16 at 19:52
Emilio Novati
51.4k43472
51.4k43472
add a comment |
add a comment |
I already see that the trivial solution $y=0$ satisfies both systems. We can slso try $y=x^3$ for the first and $y=x^2$ for the second.
answered Nov 23 at 14:23
Wesley Strik
1,486422
add a comment |
I already see that the trivial solution $y=0$ satisfies both systems. We can slso try $y=x^3$ for the first and $y=x^2$ for the second.
answered Nov 23 at 14:23
Wesley Strik
1,486422
add a comment |
I already see that the trivial solution $y=0$ satisfies both systems. We can slso try $y=x^3$ for the first and $y=x^2$ for the second.
answered Nov 23 at 14:23
Wesley Strik
1,486422
I already see that the trivial solution $y=0$ satisfies both systems. We can slso try $y=x^3$ for the first and $y=x^2$ for the second.
answered Nov 23 at 14:23
Wesley Strik
1,486422
edited Nov 23 at 14:33
answered Nov 23 at 14:23
Wesley Strik
1,486422
answered Nov 23 at 14:23
Wesley Strik
1,486422
answered Nov 23 at 14:23
Wesley Strik
1,486422
1,486422
add a comment |
add a comment |
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1
The difference is doing it in your head. You can ignore that stipulation if you want.
– arctic tern
Aug 12 '16 at 18:04