Anahi Solomon

2022-03-22

Which of the following is not a solution of $y{}^{\u2033}+4y=0$

(A)$4\mathrm{cos}\left(2x\right)$

(B)$5\mathrm{sin}\left(2x\right)$

(C)$\mathrm{sin}\left(2x\right)\mathrm{cos}\left(2x\right)$

(D)$4\mathrm{cos}\left(2x\right)+5\mathrm{sin}\left(2x\right)$

(A)

(B)

(C)

(D)

undodaonePvopxl24

Beginner2022-03-23Added 13 answers

The given equation is a second order linear non homogeneous differential equation with constant coefficient.

The general solution for$a\left(x\right){y}^{\u2033}+b\left(x\right){y}^{\prime}+c\left(x\right)y=g\left(x\right)$

The general solution of the given differential equation can be written as

$y={y}_{h}+{y}_{p}$

$y}_{h$ is the solution to the homogeneous ODE $a\left(x\right){y}^{\u2033}+b\left(x\right){y}^{\prime}+c\left(x\right)y=0$

Here the given ODE is the homogeneous equation.

The complementary solution for the given equation is:

$y{}^{\u2033}+4y=0$

Rewrite the equation with$y={e}^{\gamma \cdot x}$

$\left({e}^{\gamma \cdot x}\right){}^{\u2033}+4{e}^{\gamma \cdot x}=0$

${e}^{\gamma \cdot x}({\gamma}^{2}+4)=0$

${\gamma}^{2}+4=0\Rightarrow \text{}\gamma =2i,\text{}\gamma =-2i$

For two complex roots$\gamma}_{1}\ne q{\gamma}_{2$ , where ${\gamma}_{1}=a+i\cdot b$ , ${\gamma}_{2}=a-i\cdot b$

the general solution takes the form:$y={e}^{ax}({c}_{1}\mathrm{cos}\left(bx\right)+{c}_{2}\mathrm{sin}\left(bx\right))$

$y\left(x\right)={e}^{0}({c}_{1}\mathrm{cos}\left(2x\right)+{c}_{2}\mathrm{sin}\left(2x\right))$

$y={c}_{1}\mathrm{cos}\left(2x\right)+{c}_{2}\mathrm{sin}\left(2x\right)$

Now, there are four options and one option is incorrect:

$y={c}_{1}\mathrm{cos}\left(2x\right)+{c}_{2}\mathrm{sin}\left(2x\right)$

(A)$4\mathrm{cos}\left(2x\right)$ is possible when ${c}_{1}=4$ and ${c}_{2}=0$

(B)$5\mathrm{sin}\left(2x\right)$ is possible when ${c}_{1}=0$ and ${c}_{2}=5$

(C)$\mathrm{sin}2x\cdot \mathrm{cos}\left(2x\right)$ is not possible

(D)$4\mathrm{cos}\left(2x\right)+5\mathrm{sin}\left(2x\right)$ is possible when ${c}_{1}=4$ and ${c}_{2}=5$

Hence the solution (C) is not possible

The general solution for

The general solution of the given differential equation can be written as

Here the given ODE is the homogeneous equation.

The complementary solution for the given equation is:

Rewrite the equation with

For two complex roots

the general solution takes the form:

Now, there are four options and one option is incorrect:

(A)

(B)

(C)

(D)

Hence the solution (C) is not possible

Jeffrey Jordon

Expert2022-03-31Added 2607 answers

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