ODE4.3

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y''-y'+2y=10e^{{-x}}\sin(x)\,

Find a particular solution y_{p}\,, by finding a generalization of the equation which has the original problem as the complex part. y_{z}\, is the imaginary particular part.

(D^{2}-D+2)y_{z}=10e^{{(-1+i)x}}\,

The right hand side of the first equation is the complex part of the right hand side of the last equation. Solve for y_{z} and find the imaginary part of the answer. This will be the answer to the original problem.

y_{z}={\frac  {10e^{{(-1+i)x}}}{(-1+i)^{2}-(-1+i)+2}}\,

y_{z}={\frac  {10e^{{(-1+i)x}}}{3-3i}}={\frac  {10}{3}}{\frac  {(1+i)}{2}}e^{{-x}}(\cos(x)+i\sin(x))\,

Find the imaginary part of the last equation.

y_{p}={\mathrm  {Im}}(y_{z})={\frac  {5}{3}}e^{{-x}}(\cos(x)+\sin(x))\,

Ordinary Differential Equations

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