Differential Equations, Undetermined Coefficients - Equal Terms in

tibukooinm

tibukooinm

Answered question

2022-03-27

Differential Equations, Undetermined Coefficients - Equal Terms in the Homogeneous and Particular Solutions
y4y+3y=cosh{x}=ex+ex2

Answer & Explanation

smekkleg5hhp

smekkleg5hhp

Beginner2022-03-28Added 8 answers

Step 1
Assuming a particular solution as
yp(x)=c1(x)ex+c2(x)e3x
after substitution into the complete ODE we have
(2c1(x)+c2 (x))ex+(2c2(x)+c2 (x))e3xcosh(x)=0
now as c1(x), x2(x) are independent we follow with
{2c1(x)+c2(x)=0(2c2(x)+c2(x))e3xcosh(x)=0 
and after solving those reduced order ODE we obtain
{c1(x)=γ1e2x+γ2c2(x)=116(12e2x(1+2x+γ3))e4x+γ4 
but as we are looking for particular solutions, we choose
γ1=γ2=γ3=γ4=0
so we follow with
{c1(x)=0c2(x)=116(12e2x(1+2x))e4x 
and finally
y=c1ex+c2e3x+(116(12e2x(1+2x))e4x)e3x
or
y=(c1-18)ex+c2e3x+116e-x-14xex

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