A Non-Homogeneous Differential Equation

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PLAYLISTS 🎵 :

[leickrobinson]

I solved it a different way. Bringing y over to the LHS: y’ - y = x^2
I noticed that the derivative of ye^-x is (y’-y)e^-x. So, if we multiply both sides by e^-x, we get:
d(ye^-x)/dx = x^2 e^-x
Then we just need to integrate the RHS (e.g., using integration by parts).

[mystychief]

So funny, that the 2 different constants, both called c, do not interfere with each other in the story (by accident).

[edding]

You have such a nice voice it makes your explanations pleasant to the ears

[wesleydeng]

Can be generalized to y'= y + p(x), where p(x) is a polynomial function.

[sochjan]

The more interesting problem you obtain after replacing y with x but only on the RHS.

[sngmn]

I just used the method of variation of an arbitrary constant.

[charlesrothauser]

Use the general form, dy/dx +P(x)y = Q(x) to find u(x) = e^-x the integrating factor. Use u(x) to determine the general solution by multiplying both sides of equal sign by u(x), left side of equal sign is the derivative of the product of u(x) and y(x), integrate both sides of equal sign and simplify to find y(x).

[mechablade]

I have only studied limits and derivatives till now but still it was easy to understand this video
Thank you for making maths as entertaining as it is supposed to be

[debtanaysarkar]

Well, this is a linear differential equation, so we can solve it that way too

[UneFenetreSurLeMonde]

y(x)=a e^x + b x² + c x +d.
put y in the original equation and find a, b, c, d. Finished.

[misterdubity]

A curiosity: I took my polynomial exponents in the other direction and found f(x) = c*e^x + x^3/3 + x^4/12 + x^5/60 + ... But that is just 2*e^x minus the first 3 terms. So then I got: y = c*e^x + 2e^x - (x^2 + 2x + 2)
or if k = c+2, y = ke^x - (x^2 + 2x + 2). So I backed into the correct answer having started the polynomial towards an infinite series.

[kapsleeds]

brought me back to 2004, university 2nd year. thanks SyberMath 🤗

[Blaqjaqshellaq]

You could also say y=c*e^x - (x+1)^2 - 1.

[doge]

Isn't this just Bernoullis lijear differential equation, find the integrating factor that is e^-x and then find solution by y.e^-x = integral of x².e^-x and solve using by parts

[klementhajrullaj]

Sorry, can you help me with that exercice? Thank you! Three numbers, the sum of which is 114, can be considered as three consecutive limits of a geometric progression, or as the first, fourth and twenty-fifth limits of an arithmetical progression. Find these numbers.

[lifegamelu]

Can also be solved using the laplace transform

[kianmath]

Y = -x^2-2x-2+ce^x, linear first order deq, nice job

[TedHopp]

Unfortunate choice to use "c" for the constant in the homogeneous solution and also as the constant term for the particular solution. These should be independent values.

[leickrobinson]

Ok, this is a tough one to just do in my head, but I think the solution is… [SPOILER]



y = -x^2 - 2x - 2 + Ce^x
(whew!)

[gelbkehlchen]

Solution:
dy/dx = y+x²
————————————
Substitution: z=y+x² z’=y’+2x y’=z’-2x
—————————————
z’-2x = z |+2x ⟹
z’ = z+2x
————————————
Substitution: u=z+2x u’=z’+2 z’=u’-2
—————————————
u’-2 = u |+2 ⟹
du/dx = u+2 |*dx/(u+2) ⟹
du/(u+2) = dx |∫() ⟹
ln|u+2| = x+C |e^() ⟹
u+2 = e^(x+C) = K*e^x |Resubstitution: u=z+2x ⟹
z+2x+2 = K*e^x |Resubstitution: z=y+x² ⟹
y+x²+2x+2 = K*e^x |-x²-2x-2 ⟹
y = K*e^x-x²-2x-2

Sample:
left side: y’ = K*e^x-2x-2
right side: y+x² = K*e^x-x²-2x-2+x² = K*e^x-2x-2 ⟹
both sides are equal, all o.k.