User:Egm4313.s12.team14.kimiagarov/HW 2

Problem set 2.3 (Page59)

= Problem 3 =

Given

 * $$\displaystyle{y''+6y'+8.96y = 0}$$

Find
General solution to the differential equation

Solution
Characteristic equation:


 * $$\displaystyle{\lambda^2+6\lambda+8.96 = 0}$$

Using the quadratic equation:


 * $$\displaystyle{x = \frac{-b\plusmn\sqrt{b^2-4ac}}{2a}\,}$$

where:


 * $$\displaystyle{a=1, b=6, c=8.96}$$

Place values into quadratic equation:


 * $$\displaystyle{x = \frac{-6\plusmn\sqrt{6^2-4*1*8.96}}{2*1}\,}$$


 * $$\displaystyle{x = \frac{-6\plusmn\sqrt{36-35.84}}{2}\,}$$


 * $$\displaystyle{x = \frac{-6\plusmn0.4}{2}\,}$$


 * $$\displaystyle{x = \frac{-6.4}{2}\,,\frac{-5.6}{2}}$$


 * $$\displaystyle{x = -3.2,-2.8}$$

$$\displaystyle{y = c_1e^{-3.2x}+c_2e^{-2.8x}}$$

= Problem 4 =

Given

 * $$\displaystyle{y''+4y'+(\pi^2+4)y = 0}$$

Find
General solution to the differential equation

Solution
Characteristic equation:


 * $$\displaystyle{\lambda^2+4\lambda+(\pi^2+4) = 0}$$

Using the quadratic equation:


 * $$\displaystyle{\lambda = \frac{-b\plusmn\sqrt{b^2-4ac}}{2a}\,}$$

where:

$$\displaystyle{a=1, b=4, c=(\pi^2+4)}$$

Place values into quadratic equation:


 * $$\displaystyle{\lambda = \frac{-4\plusmn\sqrt{4^2-4*1*(\pi^2+4)}}{2*1}\,}$$


 * $$\displaystyle{\lambda = \frac{-4\plusmn\sqrt{16-4\pi^2-16)}}{2}\,}$$


 * $$\displaystyle{\lambda = \frac{-4\plusmn2\pi i}{2}\,}$$


 * $$\displaystyle{\lambda = -2\plusmn\pi i}$$

$$\displaystyle{y = e^{-2x}(C_1\cos\pi x + C_2\sin\pi x)}$$