tripes3h

2022-07-04

Finding general solution for a nonhomogeneous system of equations

$\{\begin{array}{l}{x}_{1}^{\prime}={x}_{2}+2{e}^{t}\\ {x}_{2}^{\prime}={x}_{1}+{t}^{2}\end{array}$

I want to find the general solution for it. I started by finding the general solution for the homogeneous equations:

$\left(\begin{array}{c}{x}_{1}\\ {x}_{2}\end{array}\right)={C}_{1}\left(\begin{array}{c}{e}^{t}+{e}^{-t}\\ {e}^{t}-{e}^{-t}\end{array}\right)+{C}_{2}\left(\begin{array}{c}{e}^{t}-{e}^{-t}\\ {e}^{t}+{e}^{-t}\end{array}\right)$

Now I need to find a "specific" solution for the nonhomogeneous equations but I have problems applying the method in which I make constants ${C}_{1}$ and ${C}_{2}$ a variable.

$\{\begin{array}{l}{x}_{1}^{\prime}={x}_{2}+2{e}^{t}\\ {x}_{2}^{\prime}={x}_{1}+{t}^{2}\end{array}$

I want to find the general solution for it. I started by finding the general solution for the homogeneous equations:

$\left(\begin{array}{c}{x}_{1}\\ {x}_{2}\end{array}\right)={C}_{1}\left(\begin{array}{c}{e}^{t}+{e}^{-t}\\ {e}^{t}-{e}^{-t}\end{array}\right)+{C}_{2}\left(\begin{array}{c}{e}^{t}-{e}^{-t}\\ {e}^{t}+{e}^{-t}\end{array}\right)$

Now I need to find a "specific" solution for the nonhomogeneous equations but I have problems applying the method in which I make constants ${C}_{1}$ and ${C}_{2}$ a variable.

Keegan Barry

Beginner2022-07-05Added 18 answers

System is equivalent with

${x}_{2}={x}_{1}^{\prime}-2{e}^{t},{x}_{2}^{\prime}={x}_{1}+{t}^{2}$

or

${x}_{2}={x}_{1}^{\prime}-2{e}^{t},{x}_{1}^{\u2033}={x}_{1}+{t}^{2}+2{e}^{t}.$

Firstly, by using a method of variations of constants, we will solve the second equation:

${x}_{1}^{\u2033}-{x}_{1}={t}^{2}+2{e}^{t},$

(after that, it will be easy to determine ${x}_{2}$ from the first equation). Solution of homogenuous equation is ${x}_{1}={C}_{1}{e}^{t}+{C}_{2}{e}^{-t}$, so we have to solve the following system:

${C}_{1}^{\prime}{e}^{t}+{C}_{2}^{\prime}{e}^{-t}=0,$

${C}_{1}^{\prime}{e}^{t}-{C}_{2}^{\prime}{e}^{-t}={t}^{2}+2{e}^{t},$

where ${C}_{1}$, ${C}_{2}$ are functions of $t$. It is easy to get that

${C}_{1}^{\prime}=1+\frac{{t}^{2}}{2}{e}^{-t},$

${C}_{2}(t)={C}_{2}-\frac{{e}^{2t}}{2}-\frac{{e}^{t}}{2}({t}^{2}-2t+2).$

Finally,

${x}_{1}=({C}_{1}-\frac{1}{2}){e}^{t}+{C}_{2}{e}^{-t}+t{e}^{t}-{t}^{2}-2,$

${x}_{2}=({C}_{1}-\frac{1}{2}){e}^{t}-{C}_{2}{e}^{-t}+(t-1){e}^{t}-2t.$

${x}_{2}={x}_{1}^{\prime}-2{e}^{t},{x}_{2}^{\prime}={x}_{1}+{t}^{2}$

or

${x}_{2}={x}_{1}^{\prime}-2{e}^{t},{x}_{1}^{\u2033}={x}_{1}+{t}^{2}+2{e}^{t}.$

Firstly, by using a method of variations of constants, we will solve the second equation:

${x}_{1}^{\u2033}-{x}_{1}={t}^{2}+2{e}^{t},$

(after that, it will be easy to determine ${x}_{2}$ from the first equation). Solution of homogenuous equation is ${x}_{1}={C}_{1}{e}^{t}+{C}_{2}{e}^{-t}$, so we have to solve the following system:

${C}_{1}^{\prime}{e}^{t}+{C}_{2}^{\prime}{e}^{-t}=0,$

${C}_{1}^{\prime}{e}^{t}-{C}_{2}^{\prime}{e}^{-t}={t}^{2}+2{e}^{t},$

where ${C}_{1}$, ${C}_{2}$ are functions of $t$. It is easy to get that

${C}_{1}^{\prime}=1+\frac{{t}^{2}}{2}{e}^{-t},$

${C}_{2}(t)={C}_{2}-\frac{{e}^{2t}}{2}-\frac{{e}^{t}}{2}({t}^{2}-2t+2).$

Finally,

${x}_{1}=({C}_{1}-\frac{1}{2}){e}^{t}+{C}_{2}{e}^{-t}+t{e}^{t}-{t}^{2}-2,$

${x}_{2}=({C}_{1}-\frac{1}{2}){e}^{t}-{C}_{2}{e}^{-t}+(t-1){e}^{t}-2t.$

woowheedr

Beginner2022-07-06Added 2 answers

If ${x}_{1},{x}_{2}$ are your solutions to the homogeneous equation (it doesn't actually matter what they are), look for solutions to the inhomogeneous equation in the form ${y}_{i}={v}_{i}{x}_{i}.$. If you try that, you will have simple equations for ${v}_{1},{v}_{2}.$

Find the volume V of the described solid S

A cap of a sphere with radius r and height h.

V=??

Whether each of these functions is a bijection from R to R.

a) $f(x)=-3x+4$

b) $f\left(x\right)=-3{x}^{2}+7$

c) $f(x)=\frac{x+1}{x+2}$

?

$d)f\left(x\right)={x}^{5}+1$In how many different orders can five runners finish a race if no ties are allowed???

State which of the following are linear functions?

a.$f(x)=3$

b.$g(x)=5-2x$

c.$h\left(x\right)=\frac{2}{x}+3$

d.$t(x)=5(x-2)$ Three ounces of cinnamon costs $2.40. If there are 16 ounces in 1 pound, how much does cinnamon cost per pound?

A square is also a

A)Rhombus;

B)Parallelogram;

C)Kite;

D)none of theseWhat is the order of the numbers from least to greatest.

$A=1.5\times {10}^{3}$,

$B=1.4\times {10}^{-1}$,

$C=2\times {10}^{3}$,

$D=1.4\times {10}^{-2}$Write the numerical value of $1.75\times {10}^{-3}$

Solve for y. 2y - 3 = 9

A)5;

B)4;

C)6;

D)3How to graph $y=\frac{1}{2}x-1$?

How to graph $y=2x+1$ using a table?

simplify $\sqrt{257}$

How to find the vertex of the parabola by completing the square ${x}^{2}-6x+8=y$?

There are 60 minutes in an hour. How many minutes are there in a day (24 hours)?

Write 18 thousand in scientific notation.