How to solve a differential equation with a term to a power?
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How would I solve an equation where one of the differential terms is to a power? For example:
$frac{d^2y}{dx^2}+k(frac{dy}{dx})^2=0$?
I've been given advice to use the $D$ operator which apparently means $frac{d}{dx}()$ but I'm not sure how that's applicable to this scenario. Any alternative suggestions or explanations would be appreciated!
calculus ordinary-differential-equations
New contributor
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add a comment |
$begingroup$
How would I solve an equation where one of the differential terms is to a power? For example:
$frac{d^2y}{dx^2}+k(frac{dy}{dx})^2=0$?
I've been given advice to use the $D$ operator which apparently means $frac{d}{dx}()$ but I'm not sure how that's applicable to this scenario. Any alternative suggestions or explanations would be appreciated!
calculus ordinary-differential-equations
New contributor
$endgroup$
add a comment |
$begingroup$
How would I solve an equation where one of the differential terms is to a power? For example:
$frac{d^2y}{dx^2}+k(frac{dy}{dx})^2=0$?
I've been given advice to use the $D$ operator which apparently means $frac{d}{dx}()$ but I'm not sure how that's applicable to this scenario. Any alternative suggestions or explanations would be appreciated!
calculus ordinary-differential-equations
New contributor
$endgroup$
How would I solve an equation where one of the differential terms is to a power? For example:
$frac{d^2y}{dx^2}+k(frac{dy}{dx})^2=0$?
I've been given advice to use the $D$ operator which apparently means $frac{d}{dx}()$ but I'm not sure how that's applicable to this scenario. Any alternative suggestions or explanations would be appreciated!
calculus ordinary-differential-equations
calculus ordinary-differential-equations
New contributor
New contributor
New contributor
asked Mar 29 at 18:15
Ammar TarajiaAmmar Tarajia
233
233
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1 Answer
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Since you only have second and first derivatives of $y$ and no
un-differentiated $y$, you could try to introduce the new function $v=frac{dy}{dx}$. Your differential equation will turn into $frac{dv}{dx}+kv^2=0$, and I guess you will manage to take it from here.
$endgroup$
$begingroup$
So for the substitution, could you use the chain rule? I'm fairly new to differential equations so sorry if the question sounds stupid.
$endgroup$
– Ammar Tarajia
yesterday
$begingroup$
No need for chain rule here. For the substitution, I gave all details. You now need to solve the new differential equation for $v$. Then use the result to solve for $y$.
$endgroup$
– mickep
22 hours ago
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how would I go about solving such a differential equation?
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– Ammar Tarajia
9 hours ago
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It is separable.
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– mickep
8 hours ago
add a comment |
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1 Answer
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active
oldest
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1 Answer
1
active
oldest
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active
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votes
$begingroup$
Since you only have second and first derivatives of $y$ and no
un-differentiated $y$, you could try to introduce the new function $v=frac{dy}{dx}$. Your differential equation will turn into $frac{dv}{dx}+kv^2=0$, and I guess you will manage to take it from here.
$endgroup$
$begingroup$
So for the substitution, could you use the chain rule? I'm fairly new to differential equations so sorry if the question sounds stupid.
$endgroup$
– Ammar Tarajia
yesterday
$begingroup$
No need for chain rule here. For the substitution, I gave all details. You now need to solve the new differential equation for $v$. Then use the result to solve for $y$.
$endgroup$
– mickep
22 hours ago
$begingroup$
how would I go about solving such a differential equation?
$endgroup$
– Ammar Tarajia
9 hours ago
$begingroup$
It is separable.
$endgroup$
– mickep
8 hours ago
add a comment |
$begingroup$
Since you only have second and first derivatives of $y$ and no
un-differentiated $y$, you could try to introduce the new function $v=frac{dy}{dx}$. Your differential equation will turn into $frac{dv}{dx}+kv^2=0$, and I guess you will manage to take it from here.
$endgroup$
$begingroup$
So for the substitution, could you use the chain rule? I'm fairly new to differential equations so sorry if the question sounds stupid.
$endgroup$
– Ammar Tarajia
yesterday
$begingroup$
No need for chain rule here. For the substitution, I gave all details. You now need to solve the new differential equation for $v$. Then use the result to solve for $y$.
$endgroup$
– mickep
22 hours ago
$begingroup$
how would I go about solving such a differential equation?
$endgroup$
– Ammar Tarajia
9 hours ago
$begingroup$
It is separable.
$endgroup$
– mickep
8 hours ago
add a comment |
$begingroup$
Since you only have second and first derivatives of $y$ and no
un-differentiated $y$, you could try to introduce the new function $v=frac{dy}{dx}$. Your differential equation will turn into $frac{dv}{dx}+kv^2=0$, and I guess you will manage to take it from here.
$endgroup$
Since you only have second and first derivatives of $y$ and no
un-differentiated $y$, you could try to introduce the new function $v=frac{dy}{dx}$. Your differential equation will turn into $frac{dv}{dx}+kv^2=0$, and I guess you will manage to take it from here.
answered Mar 29 at 18:22
mickepmickep
18.7k12351
18.7k12351
$begingroup$
So for the substitution, could you use the chain rule? I'm fairly new to differential equations so sorry if the question sounds stupid.
$endgroup$
– Ammar Tarajia
yesterday
$begingroup$
No need for chain rule here. For the substitution, I gave all details. You now need to solve the new differential equation for $v$. Then use the result to solve for $y$.
$endgroup$
– mickep
22 hours ago
$begingroup$
how would I go about solving such a differential equation?
$endgroup$
– Ammar Tarajia
9 hours ago
$begingroup$
It is separable.
$endgroup$
– mickep
8 hours ago
add a comment |
$begingroup$
So for the substitution, could you use the chain rule? I'm fairly new to differential equations so sorry if the question sounds stupid.
$endgroup$
– Ammar Tarajia
yesterday
$begingroup$
No need for chain rule here. For the substitution, I gave all details. You now need to solve the new differential equation for $v$. Then use the result to solve for $y$.
$endgroup$
– mickep
22 hours ago
$begingroup$
how would I go about solving such a differential equation?
$endgroup$
– Ammar Tarajia
9 hours ago
$begingroup$
It is separable.
$endgroup$
– mickep
8 hours ago
$begingroup$
So for the substitution, could you use the chain rule? I'm fairly new to differential equations so sorry if the question sounds stupid.
$endgroup$
– Ammar Tarajia
yesterday
$begingroup$
So for the substitution, could you use the chain rule? I'm fairly new to differential equations so sorry if the question sounds stupid.
$endgroup$
– Ammar Tarajia
yesterday
$begingroup$
No need for chain rule here. For the substitution, I gave all details. You now need to solve the new differential equation for $v$. Then use the result to solve for $y$.
$endgroup$
– mickep
22 hours ago
$begingroup$
No need for chain rule here. For the substitution, I gave all details. You now need to solve the new differential equation for $v$. Then use the result to solve for $y$.
$endgroup$
– mickep
22 hours ago
$begingroup$
how would I go about solving such a differential equation?
$endgroup$
– Ammar Tarajia
9 hours ago
$begingroup$
how would I go about solving such a differential equation?
$endgroup$
– Ammar Tarajia
9 hours ago
$begingroup$
It is separable.
$endgroup$
– mickep
8 hours ago
$begingroup$
It is separable.
$endgroup$
– mickep
8 hours ago
add a comment |
Ammar Tarajia is a new contributor. Be nice, and check out our Code of Conduct.
Ammar Tarajia is a new contributor. Be nice, and check out our Code of Conduct.
Ammar Tarajia is a new contributor. Be nice, and check out our Code of Conduct.
Ammar Tarajia is a new contributor. Be nice, and check out our Code of Conduct.
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