 So remember if we go remember my equation 2 here, all I've got to do now is I'm going to solve for x of 1 so that I can plug it into Equation number 1 and I'll solve for x of 2. So let's start with this. Let's start with Equation number 2 Let's get that in the picture for you. So we starting with equation number 2, which was except 2 except 2 of s and And we had s squared plus 4 minus 4 times x sub 1 of s. It was negative 1 So what we can do now is let's just solve for x of 1 So if I take that over to the other side, I'm going to have 4 times x sub 1 of s and that's going to equal x sub 2 of s and an s squared plus 4 And I'm going to have positive 1. I've taken the negative over to that side just over to this side So x of 1 of s that is going to be a quarter except 2 of s s squared plus 4 plus a quarter and let's call this Let's call this equation number 4 now I'm going to take equation number 4 and plug it back into equation number 1 and Remember what equation number 1 is? Do we still have it here somewhere? There was equation number 1 x of 1 of s is squared plus 10 minus 4 times x of 2 So I'm just going to replace the x of 1 there so that I just have x of 2's so that I can solve for x of 2's Okay, so my equation number 1 was x of 1 of s, which is now a quarter x sub 2 of s s squared plus 4 plus a quarter That was my x sub 1 of s There it still has to be multiplied by an s squared. What was it plus? It was supposed to be s squared plus 10 a negative 4 times x sub 2 of s and that had to equal That had to equal 1 Okay, let's multiply this s squared plus 10 inside of these brackets I'm going to have a quarter times x sub 2 of s s to the s squared plus 4 and an s squared plus 10 plus I'm going to have another quarter times s squared plus 10 Minus 4 times x sub 2 of s and that equals 1. I'm going to multiply throughout by 4 So I'm going to have x sub 2 of s You can multiply these out s to the power 4 plus 14 a squared Plus 40. I'm going to have an s squared plus 10 on this side That is going to be minus 16 x sub 2 of s and that's going to equal 4 So let's get x sub 2 on its own This is what we after if we do that we are left with s to the power 4 plus 14 s squared plus 40 Minus 16 just as we had before and on this side we're going to have 4 Minus s squared minus 10 Same story that we're going to have now x sub 2 of s that is going to equal we have a negative s squared Minus 6 so that's positive 6. That's what we have there We did my rule ago and in the bottom we're going to have exactly the same denominator as we had before in other words s squared plus what was a 12 and s squared plus 2 which is going to give me that 24 there Now partial fractions again, and you're going to see we're going to land up in the same same problem So let's have negative s squared plus 6 and we have Let's go through this exercise again. It's First year calculus work to do these partial fractions and It's always good exercise to do something that you used to be at least very familiar with When you were slightly younger So s squared plus That's a scale plus 2 So we're going to have a negative, you know s squared plus 6 and that is going to equal 8 times We're going to have a s squared plus 2 and b and s squared plus 12 once again There's nothing I can square to give me a zero there or zero there. I've got to invoke Is z equals s squared my complex number here and let's set the z equal to negative 2 to start off with So remember this now is z which is negative 2 plus 6 6 minus 4 is 2 Let's see the It's 4 so it's 10 so this is negative 10 Equals the a is going to fall out the b is going to be negative 2 squared as 4 as 14 14 b in other words b is going to be negative 10 over 14 Which divided by 2 Let's see if I made another silly little mistake Yeah, probably um Remember now I have to um That was a negative. That's not definitely not 10 Oh, I hate it when I'm in this kind of Brain function mode that I can't even do simple Simple simple mathematics. So that's going to become a z. Let me just do this My brain is clearly not functional this morning. So let's put a z in there and a z in there and a z in there because I'm being quite silly So if I put it if I put a negative 2 in there, that's going to be 4. So this will be a negative of 4 And if I put a negative 2 in there, that is going to be a if I put a negative 2 in there, that's going to be a 10 So what am I left with? It's negative For for b let me just see is that a negative 10 yes indeed and that has got to be a Uh 6 minus 2 is 4 negative 4. Let me just have a quick look at z plus 6 is what I did before I said it equal to negative 4. So I'm going to have negative 2 over 5 Eventually you would cry out The answer well before I get to it as you can see clearly the brain is not functional today So I've got a negative 12 there. So that gives me negative 6 that gives me a 6 Equals and on this side negative 12 negative 10. So that's negative 10 a in other words I'm going to have a equal to negative 3 over 5 Okay, so we are getting some way. So we're gonna have x sub 2 except 2 of s Is going to equal this negative Which I still have to distribute in So a why am I a was was it? No, I've already distributed it in so that's not an issue So clearly that's not an issue. So we're going to have a negative 2 over 5 over s squared plus s squared plus 2 And we're going to have a negative 3 over 5 and s squared plus 12 Eventually eventually eventually We're going to get there Now it becomes simple. We're just going to get the inverse Laplace transform on both sides Let me do this properly now. So that's x sub 2 of s That is going to be the inverse Laplace transform of Let's do that. It's going to be negative 2 over 5 over s squared plus 2 And plus the negative inverse Laplace the inverse Laplace transform of negative 3 over 5 Over s squared plus 12 That's very easy to do. This is going to become x sub 2 of t The inverse Laplace transform of that it's going to equal I'm bringing the negative 2 over 5 out I have that inverse Laplace transform of s squared plus 2 Which is just the square root of 2 squared. This is the square root of 2 squared So that I can put a square root of 2 there, which means I've got to put a 1 over square root of 2 there I don't like square root of 2 is in the denominator of an answer So I do that All in one go. Yeah, I'm going to bring the negative 3 over 5 out I am left with the inverse Laplace transform of summing Over s squared plus 12 Which is the square root of 12 squared. So that becomes the square root of 12 So that becomes 1 over. What was it? 2 square root of 3 and I have to multiply by square root of 3 over square root of 3 That leaves me x sub 2 of t Equals so this becomes 2 which goes with that. So I'm left with a negative square root of 2 over 5 here And that would be the sign of square root of 2 t. You could all recognize that This 3 this 3 and that 3 will go. So this is negative square root of 3 over 10 over 10 let's just make sure that I forgot the answer correct now seeing that Seeing that things are not too rosy in my mind this morning sign of square root of 12 is 2 square root of 3 t And beautifully I've solved my problem. It's taken me a long long time. These are long problems easy to make tiny little mistakes But so beautiful that we can use the Laplace transform then the inverse Laplace transform We use complex numbers to do our To do I didn't show you the whole method there We chucked away the the imaginary part just kept the real part in the end But I did my partial fractions there And then I had to use the inverse Laplace transform to get back To my x sub 1 and x sub 2 and I had to recognize what was left of my inverse Laplace transform to turn it into something that I could Use I had to put these new numerators there. I had to bring them out. I had to neaten things up all in all That's a very good exercise this First kind of problem to do in the morning just to get the mind going properly