 # Consider y &#x2033; </msup> + y = 2 x sin &#x2061;<!-- ⁡ --> hornejada1c 2022-07-11 Answered
Consider
${y}^{″}+y=2x\mathrm{sin}\left(x\right)$
I have the solution for the homogeneous equation. Now i am trying to guess a particular solution for: $2x\mathrm{sin}\left(x\right)$
My first guess was: $\left(Ax+B\right)\mathrm{cos}x+\left(Cx+D\right)\mathrm{sin}x$ but i end up with the system:
$\left\{\begin{array}{ccc}-2A& +2B& =0\\ 2C=0& & \end{array}$
Then my quess was: $\left(A{x}^{2}+xB\right)\mathrm{cos}x+\left(C{x}^{2}+xD\right)\mathrm{sin}x$ but that leaves me with:
$\left\{\begin{array}{cccc}-Ax& -B& +4C& =0\\ 2A& 2D& =0& \\ -4A& -Cx& -D& =0\\ -2B& 2C& =0& \end{array}$
Is here something wrong?
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The last guess is true but your calculations are incorrect.
The final equation should give.
$2A+2D=0\phantom{\rule{0ex}{0ex}}-2B+2C=0\phantom{\rule{0ex}{0ex}}-4A=2\phantom{\rule{0ex}{0ex}}4C=0$
Thus $C=B=0,A=-D=-\frac{1}{2}$