Find the general solution of the given differential equation. y” − 2y' − 3y =

Alyce Wilkinson 2021-10-23 Answered
Find the general solution of the given differential equation. \(\displaystyle{y}”−{2}{y}'−{3}{y}={3}{e}^{{{2}{t}}}\)

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Expert Answer

Jozlyn
Answered 2021-10-24 Author has 23711 answers
First find solution of homogenenous problem.
\(\displaystyle{r}^{{2}}-{2}{r}-{3}={0}\to{r}_{{{1},{2}}}=-{1},{3}\to\) using quadratic formula
Homogeneous solution:
\(\displaystyle{y}_{{c}}={c}_{{1}}{e}^{{-{t}}}+{c}_{{2}}{e}^{{{3}{t}}}\)
Let \(\displaystyle{Y}={A}{e}^{{{2}{t}}}\) because \(\displaystyle{g{{\left({t}\right)}}}={3}{e}^{{{2}{t}}}\)
Plug Y into starting equation to find particular solution:
\(\displaystyle{\left({A}{e}^{{{2}{t}}}\right)}{''}-{2}{\left({A}{e}^{{{2}{t}}}\right)}'-{3}{A}{e}^{{{2}{t}}}={3}{e}^{{{2}{t}}}\)
\(\displaystyle{4}{A}{e}^{{{2}{t}}}-{4}{A}{e}^{{{2}{t}}}-{3}{A}{e}^{{{2}{t}}}={3}{e}^{{{2}{t}}}\)
\(\displaystyle-{3}{A}{e}^{{{2}{t}}}={3}{e}^{{{2}{t}}}\)
\(\displaystyle{A}=-{1}\)
\(\displaystyle{Y}=-{e}^{{{2}{t}}}\)
Solution: \(\displaystyle{y}={y}_{{c}}={c}_{{1}}{e}^{{-{t}}}+{c}_{{2}}{e}^{{{3}{t}}}-{e}^{{{2}{t}}}\)
Result: \(\displaystyle{y}={y}_{{c}}={c}_{{1}}{e}^{{-{t}}}+{c}_{{2}}{e}^{{{3}{t}}}-{e}^{{{2}{t}}}\)
Not exactly what you’re looking for?
Ask My Question
0
 

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Relevant Questions

asked 2021-09-17
Solve each of the differential equations in Table 1. Include the characterestic polynomial and its roots with your answer.
\(\displaystyle{y}{''}-{3}{y}'+{2}{y}={0}\)
asked 2021-12-17
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.
asked 2021-12-10

The question is that find the general solution of differential equation \(\displaystyle{y}{''}-{2}{y}'+{y}={e}^{{x}}\)
i know that \(\displaystyle{y}_{{c}}={A}{x}{e}^{{x}}+{B}{e}^{{x}}\)
then let the \(\displaystyle{f{{\left({x}\right)}}}={e}^{{x}},\) so \(\displaystyle{y}={p}{x}{e}^{{x}}\) as \(\displaystyle{f{{\left({x}\right)}}}\) is in the complementary function. So \(\displaystyle{y}'={p}{e}^{{x}}{\left({x}+{1}\right)}{y}{''}={p}{e}^{{x}}{\left({x}+{2}\right)}\) then i put it into equation,but the answer is \((0)pe^x=e^x\),then i can't find the unknown number p.

asked 2021-11-19
We have the following differential equation
\(\displaystyle\cup{''}={1}+{\left({u}'\right)}^{{2}}\)
i found that the general solution of this equation is
\(\displaystyle{u}={d}{\text{cosh}{{\left({\left({x}-\frac{{b}}{{d}}\right)}\right.}}}\)
where b and d are constats
Please how we found this general solution?
asked 2021-12-31
Find the general solution of the given second-order differential equation.
3y''+2y'+y=0
asked 2021-10-02
Find the general solution of the given differential equation.
\(\displaystyle{y}{''}-{2}{y}'-{3}{y}=-{3}{t}{e}-{t}-{2}{y}'-{3}{y}=-{3}{t}{e}-{t}\)
asked 2021-11-20
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.
...