metalskaashw

2022-03-30

How to solve 2nd order inhomogeneous differential equation ivp

$\ddot{y}+4y=t-2\mathrm{sin}\left(2t\right),\text{}y\left(\pi \right)=0,\text{}\text{}\text{}\dot{y}\left(\pi \right)=1$

Pubephenedsjq

Beginner2022-03-31Added 11 answers

Step 1

The problem is that your complementary solution has the same form as the term on the right hand side. The particular solution you have tried will not work. What you can do is try

${y}_{p}=Ct\mathrm{cos}\left(2t\right)+Dt\mathrm{sin}\left(2t\right)$

Then

${\dot{y}}_{p}=C\mathrm{cos}\left(2t\right)-2Ct\mathrm{sin}\left(2t\right)+D\mathrm{sin}\left(2t\right)+2Dt\mathrm{cos}\left(2t\right)$

${\ddot{y}}_{p}=-4C\mathrm{sin}\left(2t\right)-4Ct\mathrm{cos}\left(2t\right)+4D\mathrm{cos}\left(2t\right)-4Dt\mathrm{sin}\left(2t\right)$

Notice now that

${\ddot{y}}_{p}+4{y}_{p}=-4C\mathrm{sin}\left(2t\right)+4D\mathrm{cos}\left(2t\right)$

This does not contain$t\mathrm{cos}\left(2t\right)$ or $t\mathrm{sin}\left(2t\right)$ terms.

The problem is that your complementary solution has the same form as the term on the right hand side. The particular solution you have tried will not work. What you can do is try

Then

Notice now that

This does not contain

What is the derivative of the work function?

How to use implicit differentiation to find $\frac{dy}{dx}$ given $3{x}^{2}+3{y}^{2}=2$?

How to differentiate $y=\mathrm{log}{x}^{2}$?

The solution of a differential equation y′′+3y′+2y=0 is of the form

A) ${c}_{1}{e}^{x}+{c}_{2}{e}^{2x}$

B) ${c}_{1}{e}^{-x}+{c}_{2}{e}^{3x}$

C) ${c}_{1}{e}^{-x}+{c}_{2}{e}^{-2x}$

D) ${c}_{1}{e}^{-2x}+{c}_{2}{2}^{-x}$How to find instantaneous velocity from a position vs. time graph?

How to implicitly differentiate $\sqrt{xy}=x-2y$?

What is 2xy differentiated implicitly?

How to find the sum of the infinite geometric series given $1+\frac{2}{3}+\frac{4}{9}+...$?

Look at this series: 1.5, 2.3, 3.1, 3.9, ... What number should come next?

A. 4.2

B. 4.4

C. 4.7

D. 5.1What is the derivative of $\frac{x+1}{y}$?

How to find the sum of the infinite geometric series 0.9 + 0.09 + 0.009 +…?

How to find the volume of a cone using an integral?

What is the surface area of the solid created by revolving $f\left(x\right)={e}^{2-x},x\in [1,2]$ around the x axis?

How to differentiate ${x}^{\frac{2}{3}}+{y}^{\frac{2}{3}}=4$?

The differential coefficient of $\mathrm{sec}\left({\mathrm{tan}}^{-1}\left(x\right)\right)$.