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POST SECONDARY
CALCULUS AND ANALYSIS
DIFFERENTIAL EQUATIONS
SECOND ORDER LINEAR EQUATIONS
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Linear equations questions and answers
Recent questions in Second order linear equations
Oberlaudacu
2021-12-20
Answered
If we consider the equation
\(\displaystyle{\left({1}-{x}^{{2}}\right)}{\frac{{{d}^{{2}}{y}}}{{{\left.{d}{x}\right.}^{{2}}}}}-{2}{x}{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}+{2}{y}={0},\ -{1}{ < }{x}{ < }{1}\)
how can we find the explicit solution, what should be the method for solution?
hvacwk
2021-12-19
Answered
Find equations of the tangent line and normal line to the given curve at the specified point.
\(\displaystyle{y}={2}{x}{e}^{{{x}}},\ {\left({0},\ {0}\right)}\)
Painevg
2021-12-18
Answered
Let
\(\displaystyle{y}{\left({t}\right)}\)
be a nontrivial solution for the second order differential equation
\(\displaystyle\ddot{{{x}}}+{a}{\left({t}\right)}\dot{{{x}}}+{b}{\left({t}\right)}{x}={0}\)
to determine a solution that is linearly independent from y we set
\(\displaystyle{z}{\left({t}\right)}={y}{\left({t}\right)}{v}{\left({t}\right)}\)
Show that this leads to a first order differential equation for
\(\displaystyle\dot{{{v}}}={w}\)
veksetz
2021-12-17
Answered
I need to solve the following differential equation
\(\displaystyle{x}^{{2}}{y}{''}+{\left({a}{x}-{b}\right)}{y}'-{a}{y}={0}\)
with
\(\displaystyle{a},{b}{>}{0},{x}\geq{0}\)
and
\(\displaystyle{y}{\left({0}\right)}={0}\)
. The power series method will fail since there is a singularity at x=0, while the form of the equation does not conform with the Frobenius method.
maduregimc
2021-12-16
Answered
I have this second-order ode equation:
\(\displaystyle{y}{''}-{4}{y}'+{13}{y}={0}\)
I've identified it as x missing case as
\(\displaystyle{y}{''}={f{{\left({y}',{y}\right)}}}={4}{y}'-{13}{y}\)
, so I'm substituting witth:
\(\displaystyle{y}'={P},\ {y}{''}={P}{\frac{{{\left.{d}{y}\right.}^{{2}}}}{{{d}^{{2}}{x}}}}={f{{\left({P},{y}\right)}}}={4}{P}-{13}{y}\)
At this point I have
\(\displaystyle{P}{\frac{{{d}{p}}}{{{\left.{d}{y}\right.}}}}={4}{P}-{13}{y}\)
, which seems a non-linear first-order ODE. This is currently beyond the scope of my course, so I'm unsure if I should continue and search online for solving techniques, or did I already do something wrong?
Bobbie Comstock
2021-12-16
Answered
Consider the function
\(\displaystyle{f{{\left(\mu\right)}}}=\sum_{{{i}={1}}}{\left({x}_{{{i}}}-\mu\right)}^{{{2}}},\)
where
\(\displaystyle{x}_{{{i}}}={i},\ {i}={1},\ {2},\ \cdots,\ {n}\)
What is the first and second derivative os
\(\displaystyle{f{{\left(\mu\right)}}}\)
?
Roger Smith
2021-12-10
Answered
I am trying to solve the following:
\(\displaystyle{y}{''}+{4}{y}'={\tan{{\left({t}\right)}}}\)
I have used the method of variation of parameters. Currently I am at a point in the equation where I have this:
\(\displaystyle{u}_{{1}}=\int{\frac{{{\tan{{t}}}{\cos{{2}}}{t}}}{{{2}}}}\)
I am stuck here
Donald Johnson
2021-12-08
Answered
Please suggest a substitution for solving or any method of solving:
\(\displaystyle{y}{''}{\left({x}\right)}{\cot{{\left({y}{\left({x}\right)}\right)}}}={y}'{\left({x}\right)}^{{2}}+{c}\)
tearstreakdl
2021-12-08
Answered
I got the sum of A is 0? There is no solution to this? Can someone please help. Tha
!
\(\displaystyle{y}{''}-{4}{y}'+{4}{y}=-{6}{e}^{{{2}{t}}}\)
interdicoxd
2021-12-07
Answered
Solve this differential equation:
\(\displaystyle{y}{''}-{7}{y}'+{10}{y}={\cos{{x}}}\)
where y is a one-variable function x. What am I supposed to do with the cosx?
Schwelliney
2021-11-23
Answered
I'm currently working on some problems concerning the calculus of variations and I have come up with the following differential equation that I now want to solve:
\(\displaystyle{1}+{y}'{\left({x}\right)}^{{2}}-{y}{''}{\left({x}\right)}{\left({y}{\left({x}\right)}-\lambda\right)}={0}\)
I'm only used to solving linear differential equations so I would appreciate some advice on where to start.
3kofbe
2021-11-23
Answered
Consider,
\(\displaystyle{a}{y}{''}+{b}{y}'+{c}{y}={0}\)
and
\(\displaystyle{a}\ne{0}\)
Which of the following statements are always true?
1. A unique solution exists satisfying the initial conditions
\(\displaystyle{y}{\left({0}\right)}=\pi,\ {y}'{\left({0}\right)}=\sqrt{{\pi}}\)
2. Every solution is differentiable on the interval
\(\displaystyle{\left(-\infty,\infty\right)}\)
3. If
\(\displaystyle{y}_{{1}}\)
and
\(\displaystyle{y}_{{2}}\)
are any two linearly independent solutions, then
\(\displaystyle{y}={C}_{{1}}{y}_{{1}}+{C}_{{2}}{y}_{{2}}\)
is a general solution of the equation.
alka8q7
2021-11-23
Answered
I have the following differential equation:
\(\displaystyle{y}{''}+{y}={\cos{{\left({t}\right)}}}{\cos{{\left({2}{t}\right)}}}\)
Maybe something can be done to
\(\displaystyle{\cos{{\left({t}\right)}}}{\cos{{\left({2}{t}\right)}}}\)
to make it easier to solve. Any ideas?
danrussekme
2021-11-23
Answered
Solve th second order linear equations:
\(\displaystyle{y}{''}+{4}{y}={\cos{{2}}}{x}\)
khi1la2f1qv
2021-11-21
Answered
I am having trouble figuring out a method for finding a solution to
\(\displaystyle{r}+{\left({\frac{{{2}}}{{{r}}}}\right)}{\left({r}'\right)}^{{2}}-{r}{''}={0}\)
I have tried substitution of
\(\displaystyle{w}={r}'\)
to obtain
\(\displaystyle{r}+{\left({\frac{{{2}}}{{{r}}}}\right)}{w}^{{2}}={\frac{{{d}{w}}}{{{d}{r}}}}{w}\)
But then I am not sure how to proceed from there.
Any suggestions are welcome. Tha
Edmund Adams
2021-11-21
Answered
I need to find a second order linear homogeneous equation with constant coefficients that has the given function as a solution
Queston a)
\(\displaystyle{x}{e}^{{-{3}{x}}}\)
Question b)
\(\displaystyle{e}^{{{3}{x}}}{\sin{{x}}}\)
chanyingsauu7
2021-11-21
Answered
How can the following second-order linear equation be converted into a first-order linear equation?
This is our second-order equation:
\(\displaystyle{y}{''}-{2}{y}'+{2}{y}={e}^{{{2}{t}}}{\sin{{t}}}\)
Yolanda Jorge
2021-11-20
Answered
Show that
\(\displaystyle{y}={x}^{{2}}{\sin{{\left({x}\right)}}}\)
and
\(\displaystyle{y}={0}\)
are both solutions of
\(\displaystyle{x}^{{2}}{y}{''}-{4}{x}{y}'+{\left({x}^{{2}}+{6}\right)}{y}={0}\)
and that both satisfy the conditions
\(\displaystyle{y}{\left({0}\right)}={0}\)
and
\(\displaystyle{y}'{\left({0}\right)}={0}\)
. Does this theorem contradict Theorem A? If not, why not?
Clifton Sanchez
2021-11-20
Answered
I'm trying to find the general solution to:
\(\displaystyle{y}{''}+{4}{y}={t}^{{2}}+{7}{e}^{{t}}\)
The actual problem wants me to find the initial value problem with y(0) = 0 and y'(0) = 2 but I'm confident that I can find the IVP after finding the general solution.
What I DO need help with is this:
I'm trying to set
\(\displaystyle{Y}{\left({t}\right)}={A}{t}^{{2}}+{B}{t}+{C}\)
and solving for A, B, and C for a specific solution but I find two different values for A. (1/4, and 0).
I did solve for
\(\displaystyle{e}^{{t}}\)
and found the answer to be 7/5.
Minerva Kline
2021-11-20
Is there any known method to solve such second order non-linear differential equation?
\(\displaystyle{y}{''}_{{n}}-{n}{x}{\frac{{{1}}}{{\sqrt{{{y}_{{n}}}}}}}={0}\)
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