Question

find the general solution of the homogeneous linear equations

1.(2D²-5D-3)y=0

2. (9D²-6D+1)y=0

3. (D⁴-2D²)y=0

4. (D⁵+8D³+16D)y=0

5. (D³+8)y=0

Answer 1

1.

$2k^2-5k-3=0$

$k=\frac{5\pm \sqrt{25+24}}{4}$

$k_1=-0.5,k_2=3$

$y=c_1e^{-0.5x}+c_2e^{3x}$

2.

$9k^2-6k+1=0$

$k=\frac{6\pm \sqrt{36+36}}{18}$

$k_{1,2}=1/3$

$y=c_1e^{x/3}+c_2xe^{x/3}$

3.

$x^4-2x^2=0$

$x_{1,2}=0$

$x_{3,4}=\pm \sqrt 2$

$y=c_1+c_2x+c_3e^{\sqrt2x}+c_4e^{-\sqrt2x}$

4.

$x^5+8x^3+16x=0$

$x(x^4+8x^2+16)=0$

$x^2=\frac{-8\pm \sqrt{64-64}}{2}=-4$

$x_1=0,x_{2,3}=2i,x_{4,5}=-2i$

$y=c_1+c_2cos2x+c_3sin2x+x(c_4cos2x+c_5sin2x)$

5.

$x^3+8=0$

$(x+2)(x^2-2x+4)=0$

$x_{1}=-2$

$x^2-2x+4=0$

$x_{2,3}=\frac{2\pm \sqrt{4-16}}{2}=1\pm i\sqrt 3$

$y=c_1e^{-2x}+e^x(c_2cos(x\sqrt 3)+c_3sin(x\sqrt 3))$