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Solve the differential equations.Some of the equations can be solved by the method of undetermined coefficients, but others cannot.$$y^{\prime \prime}+2 y^{\prime}=x^{2}-e^{x}$$

Calculus 3

Chapter 17

Second-Order Differential Equations

Section 2

Nonhomogeneous Linear Equations

Missouri State University

Harvey Mudd College

Baylor University

Boston College

Lectures

13:13

Solve the differential equ…

13:35

06:25

Solve the given differenti…

10:57

01:10

03:49

Use the method of undeterm…

So you have the following equation: y double pine, plus 2 y pine is equal to x, squared minus 3 x, we're going to try the method of determined coefficients, so the corresponding polynomial here for the homogeneous sliptoes to see what the male is going to be r. Squared plus 2 r equals to 0, so that can be factored r times. R plus 2 equals to 0 so that as 2 solutions are 1, is this equal to 0 r equals to 0 or r equals to minus 2 point so that how the solution to homogenous equation y sub is going to be all just a constant. It'S called a constant and then so constant d becomes e to the minus 2 x. So this is the solution for the homogeneous equation now for the particular for the solution to the particular equation. Here we see that we're taking 1 and 2 relatives. We have x, squared and nuclear x. So since we went differentiate 3 x we get again any 3 x. Any 3 x is not in the homogeneous. So let's say that they, the particular, is some constant times milo at times 3 and then we're taking at least 100 it. So the degree of the polynoe we have what we propose here should be 3, so that we be differentiated, we get 2. So we would have b x to the third power plus c x, squared dx and since the constant is already in the homogeneous run out to the homogeneous equation. So this is going to be our particular solution. You can propose that to be our particular solution, so you know that y prime of the particular would be for just a equals x with tent differentiate that we'll get plus 3 b x square plus 2 cone x, plus n. So this is y prime and the second variety is going to be the relative of this. It should be again of ganterie, 2 comes down so 3 times to so 6 d times x, plus tetti is just gone. Eton 0, so we write y 1 plus 2.5. This will be- let's write these down here, so the simulated is a x, a 83 x and 3 x, plus 6 b x, 600 x costigan. That time was that at that plus 2 times, these so 2 times a nuclex plus the b x, squared plus 2 c x, plus d, and that has to be equal to x, squared minus x, to x, squared minus 3, and to this, since these have equal. As for thomas functions is an explanation point as functions, then the coefficients were 4 x square of altona square on the side and the coefficients for wendt t 3 x is the option 3 x and the cotoneaster constant. So first coefficient is 1 with x, squared. So over this, in on this side of its going to be there's no x square here and all here- you have 2 times 3 x, squared so 62 times 3 astantes side is 1 that has 3 equal to 1. Now the coefficient that goes we x on this side is 16, o 6 plus 1 times 10 to 4 c, 6 b, plus 4 c on this side is 0 times 6, so that has to be equal to 0 and then the coefficient general with x to The 0, which is constant, that is 1, so for that we have, on this side to c to c and 2 at so 2 c, plus 2 plus 3 to all the productions here is all but an 06 to 0, so that is 0 and the we Start replacing we know from that there are these equal to 106 tustfiveeplace. What is the value of t here would have this equation and 9. There. You have 6 times oneone 6 plus 4 c is equal to 0. So the moment is the. We have that 4 c is equal to equal to minus 1 point and so c is equal to minus 1 over 4. So we have this poictiers and for b we have this equation. So 2 c, plus b is equal to 0. So, as we can like this, can reflectored like that so watson's 2 is not 0 c plus z is equal to 0 plus is equal to 0. So that means that b is equal to minus the go to minus c minus c. So with this they would be to minus minus that which i .4 is equal to. So you have the value for for for c a port, but we have to enlisting the coefficient that goes with e 3 x so have to have another equation and the pain. The coefficiency, so for that we have on this side. You have 1 on the right hand, side. So minus y will be equal to this. I have a plus 2 a so we have 3, a twonette is minus 1, and so, with those we have 8 equals to minus 1. Third, so y at the particular solution should be a times c. T t x, minus 1 for c 3 x, plus b x, cubed is 1 o 6 plus 1 over 6 x. Cube of the inner plus c x, squared and c is equal to c is 1 over 41. Over 4 c is equal to minus 1. Over 4 point so minus 1, fourth x, squared and then otocousticons that goes x is equal to 1 over 4 plus 14 point. So this is in a particular solution and therefore here there is no constant, because the 1 genus includes the cost which the 1 genus verse of the form- a plus b 83 minus minus 2 x, plus e into minus 2 x. So this is a particular homogeneous and the solution. The general solution is goot. The homogenous solution plus a particular solution which is in these proportions, and these constants are.

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