## Solving for $y” – 4y’ – 5y – 2 = 0$

I am looking to solve for the above nonhomogeneous ODE. I know how to find the general solution for the reduced equation of the homogeneous form, that is, $$y” – 4y’ – 5y = 0.$$

The characteristic equation is $$r^{2} – 4r – 5 = 0$$, which gives two real and distinct roots $$r=-1,5$$.

So the complementary solution is $$y_{c}=c_{1}e^{5x} + c_{2}e^{-x}$$.

Now I am looking to guess the particular on the right-hand side but I am not sure about how to do that in order to find the general solution of the above nonhomogenous ODE.

## Solving for Conditional Variance

I am working on a problem and am a little bit confused.

The problem:

P(X=0,Y=0) = .80

P(X=1,Y=0) = .05

P(X=0,Y=1) = .025

P(X=1,Y=1) = .125

Find Var(Y|X=1)

What I have done so far: (using)

E(Y$$^2$$|X=1) – $$\big($$E(Y|X=1)$$\big) ^2$$

But with this I have been getting:

E(Y|X=1) = $$(0)(.05) + (1)(.125)\over(.05+.125)$$ = .71

and

E(Y$$^2$$|X=1) = $$(0)^2(.05) + (1)^2(.125)\over(.05+.125)$$ = .71

But with this we get the same answer so that when we do E(X$$^2$$) – (EX)$$^2$$

We get:

.71 – .71 = 0

This doesn’t seem like it is the right answer to me?

## Solving equation with multi variable matrix input

Background, here are the equations that I am trying to solve:

Where R, E1, E2, V1, V2, P are all user inputs. X/A goes from -2 to 2 and Z/A goes from 0 to -2. Below is the code that I have so far. I created a list of inputs. Then created two arrays for the x and z inputs. The last is where I am having trouble. I’m trying to create a code such that it will hold a value for X constant in SX, SZ, and TXZ and plug in all the values for Z. Then move to the next value for X and plug all the values in for all the Z. The end goal is to create a density plot that for SX, SZ, and TXZ. Thank you!

R = .1; E1 = 200*10^9; E2 = 550*10^9; P=1000; V1 = 0.3; V2 = 0.3; E = 1/(((1-(V1^2))/E1)+((1-(V2^2))/E2)); A = ((.75*P*R)/(1.61172*10^11))^(1/3); X = Range[-2 A, 2 A, 0.01*3*A]; Z = Range[0,-2 A, 0.005*3*A];  ZZ = ConstantArray[Z[[Range[Length[Z]]]], Length[X]]; XX = ConstantArray[X[[Range[Length[X]]]],Length[Z]];  For[i=1,i=Length[XX], For[j=1,j = Length[ZZ], M = Sqrt(.5*(((A^2-i^2+j^2)^2+4*i^2*j^2)^(.5)+(A^2-i^2+j^2))) N = Sqrt(.5*(((A^2-i^2+j^2)^2+4*i^2*j^2)^(.5)-(A^2-i^2+j^2))) SX = (-P/A)*M*((1-((j^2+N^2)/(M^2+N^2)))-2*N) SZ = (-P/A)*M*((1-((j^2+N^2)/(M^2+N^2)))) SY = V1*(SX+SZ) TXZ = (-P/A)*N*((M^2-j^2)/(M^2+N^2)), DensityPlot[SX/P,XX/A,ZZ/A]  ] ]