osteoblogda

2021-12-31

Using the method of undetermined coefficients, find the general solution of the following differential equation

habbocowji

Beginner2022-01-01Added 22 answers

We find the general solution of eq (1) of the method of undesermened coefficient.

Now the aurilary eq for the homogenous eq

$y3{y}^{\prime}+2y=0$ is

${m}^{2}-3m+2=0$

$(m-2)(m-1)=0$

$m=1,2$

So the function$y={c}_{1}e+{c}_{2}{e}^{2x}$

Here we take${y}_{p}=A{x}^{2}+Bx+c$ ,

${y}_{p}=2a+b$

${y}_{p}2a$

Putting the value of$y}_{p},{y}_{p},{y}_{p$ in eq (1) we get

$2a-3(2ax+b)+2(a{x}^{2}+bx+c)={x}^{2}$

$2a-6ax-3b+2a{x}^{2}+2bx+2={x}^{2}$

$2a{x}^{2}+(-6a+2b)x+(2a-3b+2c)={x}^{2}$

$2a=1\Rightarrow a=\frac{1}{2}$

$-6A+2b=0\Rightarrow -6\times \frac{1}{2}+2b=0\Rightarrow -3+2b=0\Rightarrow b=\frac{3}{2}$

$2a-3b+2c=0\Rightarrow 2\cdot \frac{1}{2}-3\cdot \frac{3}{2}+2=0$

$\Rightarrow 1-\frac{9}{2}+2c=0$

$\Rightarrow 2c\cdot \frac{9}{2}-1$

$2c=\frac{1}{2}$

$=1c=\frac{1}{4}$

$y}_{p}=\frac{1}{2}{x}^{2}+\frac{3}{2}x+\frac{1}{4$

The general solution is

$y\left(x\right)={y}_{c}+{y}_{p}$

$={c}_{1}{e}^{x}+{c}_{2}{e}^{2x}+\frac{1}{2}{x}^{2}+\frac{3}{2}x+\frac{1}{4}$

Now the aurilary eq for the homogenous eq

So the function

Here we take

Putting the value of

The general solution is

Deufemiak7

Beginner2022-01-02Added 34 answers

Given differential equation is

$y3{y}^{\prime}+2y={x}^{2}+x+1$ (1)

$AE\text{}is\text{}{D}^{2}-3D+2=0$

$(D-1)(D-2)=0$

$\therefore D=1,2$

$\therefore CF={c}_{1}{e}^{x}+{c}_{2}{e}^{2x}$

PI is of the form

$y={A}_{0}+{A}_{1}x+{A}_{2}{x}^{2}$

${y}^{\prime}=0+{A}_{1}+2{A}_{2}x$

$y0+2{A}_{2}$

Substituting in (1)

$2{A}_{2}-3({A}_{1}+2{A}_{2}x)+2({A}_{0}+{A}_{1}x+{A}_{2}{x}^{2})={x}^{2}+x+1$

$\therefore 2{A}_{2}{x}^{2}+(2{A}_{1}-6{A}_{2})x+2{A}_{0}-3{A}_{1}+2{A}_{2}={x}^{2}+x+1$

Comparing the coefficients

$2{A}_{2}=1,2{A}_{1}-6{A}_{2}=1,2{A}_{0}-3{A}_{1}+2{A}_{2}=1$

Solving,${A}_{2}=\frac{1}{2},{A}_{1}=2,{A}_{0}=3$

$\therefore PI=3+2x+\frac{1}{2}{x}^{2}$

$\therefore GS=CF+PI$

$y={c}_{1}{e}^{x}+{c}_{2}{e}^{2x}+(3+2x+\frac{1}{2}{x}^{2})$

PI is of the form

Substituting in (1)

Comparing the coefficients

Solving,

Vasquez

Skilled2022-01-09Added 457 answers

Solve

Homogen solution:

Particular solution:

Put his into the initial equartion to get A, B and C gives me:

This leads me to the answer:

However the correct answer is

Where's my miss? Where comes the last term from?