Answer:
kckgkfjjfufufuf8f8ff88r8r8r8r8r8rrffufuufuff
The number 17 has the code 21021
68 is not a prime number. The prime factorization of 68 would be 2 x 2 x 17.
Answer:
![y(x)=3e^{5x}-2xe^{5x}](https://tex.z-dn.net/?f=y%28x%29%3D3e%5E%7B5x%7D-2xe%5E%7B5x%7D)
Step-by-step explanation:
The given differential equation is ![y''-10y'+25y=0](https://tex.z-dn.net/?f=y%27%27-10y%27%2B25y%3D0)
The characteristics equation is given by
![r^2-10r+25=0](https://tex.z-dn.net/?f=r%5E2-10r%2B25%3D0)
Finding the values of r
![r^2-5r-5r+25=0\\\\r(r-5)-5(r-5)=0\\\\(r-5)(r-5)=0\\\\r_{1,2}=5](https://tex.z-dn.net/?f=r%5E2-5r-5r%2B25%3D0%5C%5C%5C%5Cr%28r-5%29-5%28r-5%29%3D0%5C%5C%5C%5C%28r-5%29%28r-5%29%3D0%5C%5C%5C%5Cr_%7B1%2C2%7D%3D5)
We got a repeated roots. Hence, the solution of the differential equation is given by
![y(x)=c_1e^{5x}+c_2xe^{5x}...(i)](https://tex.z-dn.net/?f=y%28x%29%3Dc_1e%5E%7B5x%7D%2Bc_2xe%5E%7B5x%7D...%28i%29)
On differentiating, we get
![y'(x)=5c_1e^{5x}+5c_2xe^{5x}+c_2e^{5x}...(ii)](https://tex.z-dn.net/?f=y%27%28x%29%3D5c_1e%5E%7B5x%7D%2B5c_2xe%5E%7B5x%7D%2Bc_2e%5E%7B5x%7D...%28ii%29)
Apply the initial condition y (0)= 3 in equation (i)
![3=c_1e^{0}+0\\\\c_1=3](https://tex.z-dn.net/?f=3%3Dc_1e%5E%7B0%7D%2B0%5C%5C%5C%5Cc_1%3D3)
Now, apply the initial condition y' (0)= 13 in equation (ii)
![13=5(3)e^{0}+0+c_2e^{0}\\\\13=15+c_2\\\\c_2=-2](https://tex.z-dn.net/?f=13%3D5%283%29e%5E%7B0%7D%2B0%2Bc_2e%5E%7B0%7D%5C%5C%5C%5C13%3D15%2Bc_2%5C%5C%5C%5Cc_2%3D-2)
Therefore, the solution of the differential equation is
![y(x)=3e^{5x}-2xe^{5x}](https://tex.z-dn.net/?f=y%28x%29%3D3e%5E%7B5x%7D-2xe%5E%7B5x%7D)
Answer: (x + 1) ( x + 4)
Step-by-step explanation: