Question

find the first five terms in the soluction dy/dx + (1 + x^2)y= sinx with y(0)=a i will give a medal please

Answer 1

suppose the solution can be represented as power series
\[y = \sum\limits_{n=0}^{\infty} c_nx^n\]

\[\dfrac{dy}{dx} = \sum\limits_{n=1}^{\infty} c_n nx^{n-1}\]

plug them in the differential equation and solve the coefficients by comparing both sides

Answer 2

\[\dfrac{dy}{dx} + (1 + x^2)y= \sin x\]

\[ \sum\limits_{n=1}^{\infty} c_n nx^{n-1}+ (1 + x^2) \sum\limits_{n=0}^{\infty} c_n x^{n}= \sin x\]

replace \(\sin x\) by its power series representation \(x-\dfrac{x^3}{3!}+ \dfrac{x^5}{5!}-\cdots\)


\[ \sum\limits_{n=1}^{\infty} c_n nx^{n-1}+ (1 + x^2) \sum\limits_{n=0}^{\infty} c_n x^{n}= x-\dfrac{x^3}{3!}+ \dfrac{x^5}{5!}-\cdots\]

Answer 3

am viewing sir ride on.. please just solve to the last step. i have to study it for exam ... thank you sir

Answer 4

next expand the sums

\[ \color{purple}{\sum\limits_{n=1}^{\infty} c_n nx^{n-1}}+ \color{blue}{(1 + x^2) \sum\limits_{n=0}^{\infty} c_n x^{n}}= x-\dfrac{x^3}{3!}+ \dfrac{x^5}{5!}-\cdots\]

\[ \color{purple}{c_1+2c_2x+3c_3x^2+\cdots} + \color{blue}{(1 + x^2) (c_0+c_1x+c_2x^2+c_3x^3+\cdots )}= x-\dfrac{x^3}{3!}+ \dfrac{x^5}{5!}-\cdots\]

compare coefficients both sides

Answer 5

comparing constant terms :
\[\color{purple}{c_1}+\color{blue}{c_0} = 0\tag{1}\]

comparing x coefficients :
\[\color{purple}{2c_2}+\color{blue}{c_1} = 1\tag{2}\]

comparing x^2 coefficients :
\[\color{purple}{3c_3}+\color{blue}{c_2+c_0} = 0\tag{3}\]

Answer 6

From the given initial condition \(y(0) = a \implies c_0 = a\)

plug this in equation \((1)\) and obtain \(c_1 = -a\)

plug this in equation \((2)\) and obtain \(c_2 = \dfrac{a+1}{2}\)

plug these in equation \((3)\) and obtain \(c_3 = -\dfrac{3a+1}{6}\)

so the solution would be :
\[y = \sum\limits_{n=0}^{\infty} c_nx^n = a -ax + \frac{a+1}{2}x^2 -\frac{3a+1}{6}x^3 + \cdots\]

Answer 7

Thanks for every thing sir. I really appreciate. Please I have another question
Find the two linear independent soluction of
D^2y/dx^2 + x day/dx + y = 0 where the series expansion is about the origin

Answer 8

Please help sir