Question

Solve the following ordinary differential equations
d^2y/d x^2 +d y/d x+y=0​

Answer 1

Step-by-step explanation:

This is a 2nd Order Homogeneous ODE below

d2ydx2+2dydx+y=0

At x=0,y=1,dydx=2

I often write (0,1,2) as a shorthand. Firstly we set up our auxiliary equation by replacing each differential with a variable, lets say m such that:

m2+2m+1=0

Solving this yields

m=−1

Therefore this is a repeated root solution yielding a general solution of the following format

y=Ae−x+Bxe−x

Notice how the coefficient of the exponent is equal to the root of the auxiliary equation.

Now we can substitute in the values of x and y in an attempt to find A,B

at(0,1),1=A⟹A=1

Now to find B we must differentiate this and plug in the (x,dydx) values

dydx=−Ae−x−Bxe−x+Be−x

at(0,2),2=−1+B⟹B=3

Now we can put together our particular solution

y=e−x+3xe−x

Simplifying yields

y=e−x(3x+1)

This is the particular solution that satisfies the above differential equation

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