Find the general solution for the nonhomogeneous differential equation:

{eq}y" + 3y' - 10y = 6e^{2x} {/eq}

Question:

Find the general solution for the nonhomogeneous differential equation:

{eq}y" + 3y' - 10y = 6e^{2x} {/eq}

Linear Equations with Constant Coefficient:


A linear equation is one in which the dependent variable {eq}y{/eq} and is derivative of any order occurs only in the first degree and are not multiplied together, their coefficients being constants or functions of the independent variable {eq}x{/eq}.


Answer and Explanation: 1


Given:


  • Consider the given non-homogeneous differential equation {eq}y'' + 3y' - 10y = 6{e^{2x}}{/eq} .


Let us now solve the above differential equation.


Let {eq}D = \dfrac{d}{{dx}},{D^2} = \dfrac{{{d^2}}}{{dx}}{/eq} . by plugging this in the given equation, we get,


{eq}\left( {{D^2} + 3D - 10} \right)y = 6{e^{2x}}{/eq}


First, we shall find the complementary function as follows,


Let {eq}D = m{/eq}, then to find the complementary function, consider


{eq}{m^2} + 3m - 10 = 0{/eq}


By factorizing we get,


{eq}m = - 5,2{/eq}


Therefore the complementary function is


{eq}{y_1} = A{e^{ - 5x}} + B{e^{2x}}{/eq}


Now let us find the particular integral for {eq}6{e^{2x}}{/eq} .


As {eq}f\left( a \right) = 0{/eq} we find that the particular integral is of the form {eq}Cx{e^{2x}}{/eq} , {eq}C{/eq} is a constant.


Let us now find {eq}C{/eq},


Consider,


{eq}{y_2} = Cx{e^{2x}}{/eq}


Differentiating with respect to {eq}x{/eq} , we get


{eq}{y_2}' = 2Cx{e^{2x}} + C{e^{2x}}{/eq}


Again differentiating,


{eq}{y_2}'' = 4Cx{e^{2x}} + 4C{e^{2x}}{/eq}


Substituting these values in the given equation we get,


{eq}\begin{align*} \left( {4Cx{e^{2x}} + 4C{e^{2x}}} \right) + 3\left( {2Cx{e^{2x}} + C{e^{2x}}} \right) - 10Cx{e^{2x}} &= 6{e^{2x}}\\ 7C{e^{2x}} &= 6{e^{2x}}\\ C &= \dfrac{6}{7} \end{align*}{/eq}


Thus,


{eq}{y_2} = \dfrac{6}{7}x{e^{2x}}{/eq}


Therefore the solution is {eq}y = {y_1} + {y_2}{/eq}


Hence, {eq}y = A{e^{ - 5x}} + {e^{2x}}\left( {B + \dfrac{6}{7}} \right){/eq} .


Learn more about this topic:

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Solving Systems of Linear Differential Equations by Elimination

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Chapter 13 / Lesson 7
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Differential equations are equations where rates of change occur with respect to variables. Learn how to solve systems of linear differential equations by elimination, using a step-by-step example to reduce the system to one equation.


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