00:01
In first a part, the given differential equation is y double dash plus 5 y -dice plus 6 y is equal to 0.
00:11
At the point of solution.
00:12
As we can see that the given differential equation is homogeneous.
00:16
And the solution will be y is equal to complementary solution.
00:19
For complementary solution, its auxiliary equation is m square plus 5m plus 6 is equal to 0.
00:28
So from here m square plus 5 of m plus thrice of m plus 3m plus 6 equal to 0.
00:37
So it can affect rise m is am multiplied to m plus 2 3 common m plus 2 is equal to 0 so from here m plus 3 multiply to m plus 2 is equal to 0 so this implies m is equal to negative 2 and negative 3, hence this solution will be y of x equal to c1, it is by negative price of x plus d2, it does by negative price of x.
01:09
So this is our required complementary solution.
01:20
Now, in part 2, the given differential equation is y double -dice plus 5y -dice plus 6 y is equal to j is the condition y of 0 is equal to y dies of 0 is equal to so it's a homogeneous and solution be free complementary solution now it's the part from sorry from part one its solution is y of x equal to c1 e to b negative y of x plus c2 a 2x2x0.
02:03
Now differentiate y -dice of 0.
02:06
Y -dice of x is equal to negative y -s of c1, 2 to 2 2 to the bar negative twice of x plus negative dice of c2 2 to the condition y of 0 2 is implies c1 plus c2 is equal to 2...