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Solve by using the method of undetermined coefficients. π‘₯ 2 𝑑 2𝑦 𝑑π‘₯ 2 βˆ’ 3π‘₯ 𝑑𝑦 𝑑π‘₯ + 5𝑦 = π‘₯ 2 sin π‘™π‘œπ‘”π‘₯

Solve by using the method of undetermined coefficients. π‘₯ 2 𝑑 2𝑦 𝑑π‘₯ 2 βˆ’ 3π‘₯ 𝑑𝑦 𝑑π‘₯ + 5𝑦 = π‘₯ 2 sin π‘™π‘œπ‘”π‘₯Β 

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Grade:12th pass

1 Answers

Harshit Singh
askIITians Faculty 5964 Points
6 months ago
Dear Student

Given Differential equation isπ‘₯2𝑦″−3π‘₯𝑦′+5𝑦=π‘₯2𝑠𝑖𝑛(π‘™π‘œπ‘”π‘₯).
x2y″−3xy′+5y=x2sin(logx).
Since it is second order linear differential equation (Euler - Cauchy equation),
the homogeneous part of equation can be solved as follows,

Let𝑦=π‘₯π‘Ÿ,y=xr,to get the characteristic equation, which is

π‘Ÿ(π‘Ÿ−1)−3π‘Ÿ+5=π‘Ÿ2−4π‘Ÿ+5=0r(r−1)−3r+5
=r2−4r+5=0

π‘Ÿ=2+𝑖,2−𝑖.r=2+i,2−i.

Thus the homogeneous solutionπ‘¦β„Ž=𝑐1π‘₯2𝑠𝑖𝑛(π‘™π‘œπ‘”π‘₯)+𝑐2π‘₯2π‘π‘œπ‘ (π‘™π‘œπ‘”π‘₯).yh=c1x2sin(logx)+c2x2cos(logx).

Now the particular solution is found by many ways, here I will just guess the particular solution to beπ‘Žπ‘₯2𝑙𝑛(π‘₯)π‘π‘œπ‘ (π‘™π‘œπ‘”π‘₯).ax2ln(x)cos(logx).

To find the value of the constantπ‘Ž,a,we substitute the the particular solution into differential equation.

π‘Žπ‘₯2((π‘™π‘œπ‘”π‘₯+3)π‘π‘œπ‘ (π‘™π‘œπ‘”π‘₯)−(3π‘™π‘œπ‘”π‘₯+2)𝑠𝑖𝑛(π‘™π‘œπ‘”π‘₯))+π‘Žπ‘₯2((2π‘™π‘œπ‘”(π‘₯)+1)π‘π‘œπ‘ (π‘™π‘œπ‘”π‘₯)−π‘™π‘œπ‘”π‘₯𝑠𝑖𝑛(π‘™π‘œπ‘”π‘₯))−4π‘Žπ‘₯2𝑙𝑛π‘₯π‘π‘œπ‘ (π‘™π‘œπ‘”π‘₯)=−2π‘Žπ‘₯2𝑠𝑖𝑛(π‘™π‘œπ‘”π‘₯)=π‘₯2𝑠𝑖𝑛(π‘™π‘œπ‘”π‘₯)ax2((logx+3)cos(logx)−(3logx+2)sin(logx))+ax2((2log(x)+1)cos(logx)−logxsin(logx))−4ax2lnxcos(logx)=−2ax2sin(logx)=x2sin(logx)

Soπ‘Ž=−12.a=−12.

Thus the solution is𝑦=𝑐1π‘₯2𝑠𝑖𝑛(π‘™π‘œπ‘”π‘₯)+𝑐2π‘₯2π‘π‘œπ‘ (π‘™π‘œπ‘”π‘₯)−12π‘₯2𝑙𝑛(π‘₯)π‘π‘œπ‘ (π‘™π‘œπ‘”π‘₯).

Thanks

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