Determine the form of a particular solution to L(y)=phi (x) for phi(x) as given if the solution to the associated homogeneous equation L(y)=0 is y_h=c_1 e^(2x)+c_2 e^(3x) 1) phi(x)=2x-7 2) phi(x)=-3x^2 3) phi(x)=4e^(2x) 4) phi(x)=2 cos (3x)

Determine the form of a particular solution to $L\left(y\right)=\varphi \left(x\right)$ for $\varphi \left(x\right)$ as given if the solution to the associated homogeneous equation L(y)=0 is ${y}_{h}={c}_{1}{e}^{2x}+{c}_{2}{e}^{3x}$
$1\right)\varphi \left(x\right)=2x-7\phantom{\rule{0ex}{0ex}}2\right)\varphi \left(x\right)=-3{x}^{2}\phantom{\rule{0ex}{0ex}}3\right)\varphi \left(x\right)=4{e}^{2x}\phantom{\rule{0ex}{0ex}}4\right)\varphi \left(x\right)=2\mathrm{cos}\left(3x\right)$
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

fairymischiefv9
Consider the differential equation
$L\left[y\right]=\varphi \left(x\right)$
The complementary solution associated with L[y] is ${y}_{h}={c}_{1}{e}^{2x}+{c}_{2}{e}^{3x}$
1)Find the form of particularsolution if This function is polynomial and no repetition of terms with ${y}_{h}$ , the form of particular solution is ${y}_{p}\left(x\right)=Ax+B$
2)Find the form of particularsolutionif $\varphi \left(x\right)=-3{x}^{2}$This function is polynomial and no repetition of terms with ${y}_{h}$ , the form of particular solution is ${y}_{p}\left(x\right)=A{x}^{2}+Bx+C$
3)Find the form of particularsolution if $\varphi \left(x\right)=4{e}^{2x}$This function is exponential and thereis repetition of terms with ${y}_{h}\left({c}_{1}{e}^{2x}\right)$, the form of particular solution is ${y}_{p}\left(x\right)=xA{e}^{2x}$ ,(if there is no repetition then the particular solution form is ${y}_{p}\left(x\right)=A{e}^{2x}$
4)Find the form of particular solution if $\varphi \left(x\right)=2\mathrm{cos}3x$This function is trigonometric and there is no repetition of terms with ${y}_{h}$ , the form of particular solution is ${y}_{p}\left(x\right)=A\mathrm{cos}3x+B\mathrm{sin}3x$