State the general solution to the differential equation,...

State the general solution to the differential equation, given that $y_1$ is a solution. (Use $c_1$ and $c_2$ as the coefficients of the solutions $y_1(x)$ and $y_2(x)$, respectively.) y'' - 3y' - 4y = 0, $y_1(x) = e^{4x}$ y(x) = Question Help: Post to forum

Transcript

00:01 Here in this question it is given that y double dash negative to y -dash negative 2y dash 3y is equals to x e to the power to x here we need to find solution of differential equation using undetermined coefficient so here here consider the given differential equation as y double dash negative to y -dash negative 2y -nogative x e -dh to the power 2x.

01:18 Now here the auxiliary equation that is ae will be equals to m square negative 2m plus 3 is equals to 0 here it will be written as m square negative 3m plus m and here we have negative negative negative 3 is equal to 0 now here we can take m common so here we have m negative 3 plus here we have 1 which we can take common from this one here we have m negative 3 which is equals to 0 now from the whole we can take m negative 3 common and here we have m plus 1 now when we equate both of them to equals to 0 we get m negative 3 is equals to 0 and m plus 1 is equal to 0 so this implies that value of m is equals to 3 and this implies the value of m is equals to negative 1 so here we have complementary function that is y c is equals to c1 e to the power 3x plus c2 e to the power negative x now here for the particular solution we let y p is equals to a x e dash to the power 2x so here we evaluate y p dash that will be equals to a e to the power 2x plus a 2 e to the power 2x and here we have x now we evaluate y p double dash so here it will be equals to 2a e r s to the power 2x plus 2a e raise to the power 2x plus 4a x x e raise to the power 2x plus 4a x e we'll put this in the expression that is y double dash negative to y dash negative 2y dash 3y is equals 2 x e dash to the power 2x now here we'll apply p now here on putting up these three values in this expression here we have 2a e dash to the power 2x plus 2a e to the power 2x plus 4 a x e to the power 2x negative a e the power 2x negative here we have 4 a x e to the power 2x negative 3a x e to the power 2x which is equals to x e to the power 2x now here here we can see that from this term this term will get cancelled out also from this term and this term will get cancelled out and here we have 2a e -rish to the power 2x negative 3a x e -res to the power 2x is equals to x x e -dish to the power 2x.

05:10 Now here we can take a, e, dash to the power 2x common, and here we have 2 negative 3x which is equals to x, e reach to the power 2x...

Question

State the general solution to the differential equation, given that $y_1$ is a solution. (Use $c_1$ and $c_2$ as the coefficients of the solutions $y_1(x)$ and $y_2(x)$, respectively.) y'' - 3y' - 4y = 0, $y_1(x) = e^{4x}$ y(x) = Question Help: Post to forum

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