Second order differential equation using double substitution

acidkc111

1

x(d^2y/dx^2) + x(dy/dx)^2 + 0.5(dy/dx)=0.25 hint use x=t^2 , u=e^y then derive a differential equation for u(t)

Reply 1

TeeEm

19

Original post by acidkc111

x(d^2y/dx^2) + x(dy/dx)^2 + 0.5(dy/dx)=0.25 hint use x=t^2 , u=e^y then derive a differential equation for u(t)

are you posing this as a challenge?

Reply 2

acidkc111

OP

1

I completed all my work and I asked my professor for a bonus question. Just a hint on how you'd go about solving with the substitution would help

Reply 3

TeeEm

19

Original post by acidkc111

I completed all my work and I asked my professor for a bonus question. Just a hint on how you'd go about solving with the substitution would help

the hint given is sufficient

use the first substitution first then the second
(However this is not a sensible way of solving this ... I agree with the first substitution then I would use something which very obvious)

Reply 4

acidkc111

OP

1

Original post by TeeEm

the hint given is sufficient

use the first substitution first then the second
(However this is not a sensible way of solving this ... I agree with the first substitution then I would use something which very obvious)

would you differentiate x=2t with respect to u so (dx/du)=2(dt/du) then have (dy/dx)=(dy/du)*(du/dx) where (dy/du)=e^-y so (dy/dx)=(1/2)*(e^-y)(du/dt)

Reply 5

TeeEm

19

Have you covered substitutions when it comes to ODEs?

Reply 6

acidkc111

OP

1

yes but im not sure what to do when I am given 2 hints it confuses me more than anything

Reply 7

acidkc111

OP

1

i used the substitution x=t^2 and got (dx/dt)=2t (dy/dx)=(dy/dt)(dt/dx) (dy/dx)=(dy/dt)(1/2t) (d^2y/dx^2)=((d^2y/dt^2)*1/2t -1/2t^2(dy/dt))1/2t subbing this in and cancelling I got

d^2y/dt^2 +(dy/dt)^2=1

Reply 8

TeeEm

19

Original post by acidkc111

i used the substitution x=t^2 and got (dx/dt)=2t (dy/dx)=(dy/dt)(dt/dx) (dy/dx)=(dy/dt)(1/2t) (d^2y/dx^2)=((d^2y/dt^2)*1/2t -1/2t^2(dy/dt))1/2t subbing this in and cancelling I got

d^2y/dt^2 +(dy/dt)^2=1

correct

now use the other substitution (easier)

Reply 9

acidkc111

OP

1

Original post by TeeEm

correct

now use the other substitution (easier)

Should I use the substitution from where I am at now or from the beginning and what should I differentiate u=e^y with respect to

Reply 10

TeeEm

19

Original post by acidkc111

Should I use the substitution from where I am at now or from the beginning and what should I differentiate u=e^y with respect to

from where you are now

Reply 11

acidkc111

OP

1

I wrote u=e^y as lnu=y -> 1/u = dy/du dy/dt = (u^-1 du/dt) d^2y/dt^2 =(-u^-2)(du/dt)^2 +(u^-1)(d^2u/dt^2) subbing in

d^2u/dt^2 -u=0 the Aux equation is m^2-1=0 m=1 m=-1 so u(t)=Ae^t+Be^-t

Second order differential equation using double substitution

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