How to find particular solution of differential equation

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Find a particular solution for the differential equation by the method of undetermined coefficients. $$2y'' - 16y' + 32y = -e^{4x}$$ Also, find the general solution of this equation.

The steps I took to solve this problem,

Find the auxiliary equation which is $2m^2-16m+32=0$ for which the roots are $m_1=4$ and $m_2=4$ so $m=4$ of multiplicity 2.

Solve for a general equation of $y_h(x) = C_1e^{4x}x + C_2e^{4x}$

When I try to find a particular solution by taking the derivates of the right hand side, I get \begin{align} y_p &= Ae^{4x}\\ y_p' &= 4Ae^{4x}\\ y_p'' &= 16Ae^{4x} \end{align} Substituting these values into the left hand side results in $0 = -e^{4x}$ which is not possible. Can someone identify what I am missing?

Pragabhava

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asked Nov 20, 2012 at 5:57

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Your problem is $e^{4x}$ is part of your homogeneous solution, which explains why you get $0$ when you try $Ae^{4x}$ as a particular solution.

I would try $Ax^2e^{4x}$ as a particular solution.

Just to show it works, let me show you another way to solve the problem. Let

$$z=y'-4y,z'=y''-4y'$$ $$y''-8y'+16y=-\frac12e^{4x}$$ $$(y'-4y)'-4(y'-4y)=z'-4z=-\frac12e^{4x}$$ $$e^{-4x}z'-4e^{-4x}z=(e^{-4x}z)'=-\frac12$$ $$e^{-4x}z=-\frac12x+k_1,z=-\frac12xe^{4x}+k_1e^{4x}$$ $$y'-4y=-\frac12xe^{4x}+k_1e^{4x}$$ $$e^{-4x}y'-4e^{-4x}y=(e^{-4x}y)'=-\frac12x+k_1$$ $$e^{-4x}y=-\frac14x^2+k_1x+k_2,y=-\frac14x^2e^{4x}+k_1xe^{4x}+k_2e^{4x}$$

So our particular solution turns out to be $-\frac14x^2e^{4x}$

answered Nov 20, 2012 at 6:07

MikeMike

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In case of a double root, the method fails, we need to start over with a different assumption of the solution. This time though, we approach the problem in a more informed manner.

hint: when we solve a simple homogeneous system, when we have a double root, to obtain a second LI solution we simply multiply root by x.

answered Nov 20, 2012 at 6:07

Aseem DuaAseem Dua

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To find the particular solution, you simply take your general solution and plug in the values that you are given for the particular solution.

Your general solution is

$$y=Ae^x+Be^{2x}+2\sin x+6\cos x.$$

You have given that the particular solution has the properties $y(0)=0$ and $\frac{dy}{dx}(0)=0$. The first condition means that when $x=0$, then $y=0$, so plug $x=0$ into your general solution and set it equal to $0$:

$$0 = A+B+6.$$

To use the second condition, you first need to differentiate your general solution:

$$\frac{dy}{dx}=Ae^x+2Be^{2x} + 2\cos(x) - 6\sin(x),$$

and then the second condition says that when $x=0$, then $\frac{dy}{dx}=0$, so you can plug in $x=0$, and set it equal to $0$:

$$0 = A+2B+2.$$

What you have now is two equations in the two unknowns $A$ and $B$, which you can solve. Take the resulting values for $A$ and $B$ and plug them back into your general solution: You now have the particular solution you were looking for.

Particular solution of the differential equation is a unique solution of the form y = f(x), which satisfies the differential equation. The particular solution of the differential equation is derived by assigning values to the arbitrary constants of the general solution of the differential equation.

Let us learn more about the particular solution of the differential equation, how to find the solution of the differential equation, and the difference between a particular solution and the general solution of the differential equation.

What Is Particular Solution Of The Differential Equation?

Particular solution of the differential equation is an equation of the form y = f(x), which do not contain any arbitrary constants, and it satisfies the differential equation. The equation or a function of the form y = f(x), having specific values of x which satisfy this equation and are called the solutions of this equation. For a differential equation d2y/dx2 + 2dy/dx + y = 0, the the values of y which satisfy this differential equation is called the solution of the differential equation.

Here y = f(x) representing a line or a curve is the solution of the differential equation that satisfies the differential equation. The solution of the form y = ax2 + bx + c is the general solution of the differential equation, since it contains arbitrary constants a, b, c. Further, if the solution has values assigned to these arbitrary constants, or if the solution is without any arbitrary constants, then the solution is called the particular solution of the differential equation.

How To Find Particular Solution Of Differential Equation?

The particular solution of the differential equation can be computed from the general solution of the differential equation. The general solution of a differential solution would be of the form y = f(x) which could be any of the parallel line or a curve, and by identifying a point that satisfies one of these lines or curves, we can find the exact equation of the form y = f(x) which is the particular solution of the differential equation.

The following steps help in finding the particular solution of the differential equation.

• The given differential equation is solved by separating the variables and integrating on both sides to obtain the general solution of the differential equation.
• For differential equations that cannot be solved easily, different methods are employed to find the general solution of the differential equation.
• The general solution of the differential equation contains arbitrary constants, which have to be assigned suitable values to get a particular solution of the differential equation.
• A point is identified to help substitute the values for the arbitrary constants, to obtain the particular solution of the differential equation.
• There can be more than one particular solution for the differential equation, based on the different values of the arbitrary constant.

Particular Solution Vs General Solution Of Differential Equation

A particular solution of the differential equation is derived from the general solution of the differential equation. The differential equation has one general solution, and numerous particular solutions, based on the different values of the arbitrary constants of the general solution.

The general solution of the differential represents a family of curves or lines in the coordinate plane, These curves or lines represent a set of parallel lines or curves, and each of these lines or the curves can be identified as the particular solution of the differential equation.

The general solution of the differential equation is of the form y = ax + b, but the particular solution of the differential equation can be y = 3x + 4, y = 5x + 7, y = 2x + 1. These particular solutions of the differential equation have been obtained by assigning different values to the arbitrary constants a, b in the general solution of the differential equation.

Related Topics

The following topics will help in a better understanding of the particular solution of the differential equation.

• Order and Degree of Differential Equation
• Linear Differential Equation
• Differential Equations
• Homogeneous Differential Equation

FAQs on Particular Solution Of The Differential Equation

What Is The Particular Solution Of The Differential Equation?

A particular solution of differential equation is a solution of the form y = f(x), which do not have any arbitrary constants. The general solution of the differential equation is of the form y = f(x) or y = ax + b and it has a, b as its arbitrary constants. Attributing values to these arbitrary constants results in the particular solutions such as y = 2x + 1, y = 3x + 4, y = 5x + 2.

What Is The Difference Between General Solution And Particular Solution Of Differential Equation?

The particular solutions have been derived from the general solutions. The general solution of a differential equation has arbitrary constants, and the solutions without any arbitrary constants is called the particular solution of the differential equation. The general solution of the differential equation represents a family of curves or a set of parallel lines, and each of these lines or curves can be terms as the particular solution of differential equation.

How Do You Identify A Particular Solution Of Differential Equation?

The particular solution of differential equation can be easily identified, as it does not have any arbitrary constants. The solutions y = 3x + 3, y = x2 + 11x + 7, are the examples of particular solution of differential equation.

What Is The Use Of a Particular Solution Of Differential Equation?

The particular solution of differential equation is useful to find the exact solution satisfying the differential equation, at a particular point or for a particular value of the independent variable.