How to solve 2nd order inhomogeneous differential equation ivpy¨+4y=t−2sin⁡(2t), y(π)=0, y˙(π)=1

Step 1The 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 tryyp=Ctcos⁡(2t)+Dtsin⁡(2t)Theny˙p=Ccos⁡(2t)−2Ctsin⁡(2t)+Dsin⁡(2t)+2Dtcos⁡(2t)y¨p=−4Csin⁡(2t)−4Ctcos⁡(2t)+4Dcos⁡(2t)−4Dtsin⁡(2t)Notice now thaty¨p+4yp=−4Csin⁡(2t)+4Dcos⁡(2t)This does not contain tcos⁡(2t) or tsin⁡(2t) terms.

Explained: "How to solve 2nd order inhomogeneous differential equation..."

High School
Calculus and Analysis
Calculus 2
Differential equations

metalskaashw

Answered question

2022-03-30

How to solve 2nd order inhomogeneous differential equation ivp

\ddot{y} + 4 y = t - 2 \sin (2 t), y (π) = 0, \dot{y} (π) = 1

Answer & Explanation

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} = C t \cos (2 t) + D t \sin (2 t)

Then

{\dot{y}}_{p} = C \cos (2 t) - 2 C t \sin (2 t) + D \sin (2 t) + 2 D t \cos (2 t)

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

Notice now that

{\ddot{y}}_{p} + 4 y_{p} = - 4 C \sin (2 t) + 4 D \cos (2 t)

This does not contain

t \cos (2 t)

t \sin (2 t)

terms.

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Calculus 2

What is the derivative of the work function?
How to use implicit differentiation to find $\frac{d y}{d x}$ given $3 x^{2} + 3 y^{2} = 2$ ?
How to differentiate $y = \log x^{2}$ ?
The solution of a differential equation y′′+3y′+2y=0 is of the form
A) $c_{1} e^{x} + c_{2} e^{2 x}$
B) $c_{1} e^{- x} + c_{2} e^{3 x}$
C) $c_{1} e^{- x} + c_{2} e^{- 2 x}$
D) $c_{1} e^{- 2 x} + c_{2} 2^{- x}$
How to find instantaneous velocity from a position vs. time graph?
How to implicitly differentiate $\sqrt{x y} = x - 2 y$ ?
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. 5.1
What 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 $f (x) = e^{2 - x}, x \in [1, 2]$ around the x axis?
How to differentiate $x^{\frac{2}{3}} + y^{\frac{2}{3}} = 4$ ?
The differential coefficient of $\sec (\tan^{- 1} (x))$ .

Didn't find what you were looking for?