The Corresponding Homogeneous Equation

Oct 13, 2025 by miodeya.com 74 views

Introduction

In this article, we will focus on finding the general solution of the given differential equation $y^{\prime \prime}-2 y^{\prime}-3 y=-3 t e^{-t}$ . This equation is a second-order linear non-homogeneous differential equation with constant coefficients. The general solution of such an equation consists of two parts: the complementary solution and the particular solution. The complementary solution is the solution of the corresponding homogeneous equation, while the particular solution is a specific solution that satisfies the non-homogeneous equation.

The Corresponding Homogeneous Equation

The corresponding homogeneous equation of the given differential equation is $y^{\prime \prime}-2 y^{\prime}-3 y=0$ . To find the complementary solution, we need to solve this equation.

Finding the Complementary Solution

The characteristic equation of the homogeneous equation is $r^2-2r-3=0$ . We can solve this quadratic equation using the quadratic formula: $r=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ . In this case, $a=1$ , $b=-2$ , and $c=-3$ . Plugging these values into the quadratic formula, we get:

r=\frac{2\pm\sqrt{(-2)^2-4(1)(-3)}}{2(1)}=\frac{2\pm\sqrt{16}}{2}=\frac{2\pm4}{2}

This gives us two possible values for $r$ : $r_1=3$ and $r_2=-1$ . Since the characteristic equation has two distinct real roots, the complementary solution is of the form $y_c=c_1 e^{r_1 t}+c_2 e^{r_2 t}$ .

Finding the Particular Solution

To find the particular solution, we need to find a solution that satisfies the non-homogeneous equation. In this case, the non-homogeneous term is $-3 t e^{-t}$ . We can try to find a solution of the form $y_p=A t e^{-t}$ , where $A$ is a constant.

Verifying the Particular Solution

To verify that $y_p=A t e^{-t}$ is a solution, we need to plug it into the original differential equation and check if it satisfies the equation. We have:

y^{\prime \prime}-2 y^{\prime}-3 y=-3 t e^{-t}

Plugging in $y_p=A t e^{-t}$ , we get:

y^{\prime \prime}_p-2 y^{\prime}_p-3 y_p=-3 t e^{-t}

Using the product rule for differentiation, we get:

y^{\prime \prime}_p=(A e^{-t}-A t e^{-t})^{\prime \prime}=(A e^{-t}-A t e^{-t})^{\prime}(-e^{-t})+A e^{-t}(-e^{-t})

Simplifying, we get:

y^{\prime \prime}_p=(-2 A e^{-t}+A t e^{-t})(-e^{-t})+A e^{-t}(-e^{-t})

y^{\prime \prime}_p=2 A e^{-t}-A t e^{-t}-2 A e^{-t}+A t e^{-t}

y^{\prime \prime}_p=0

Similarly, we can find $y^{\prime}_p$ :

y^{\prime}_p=(A e^{-t}-A t e^{-t})^{\prime}=A e^{-t}-A t e^{-t}-A e^{-t}

y^{\prime}_p=-A t e^{-t}

Plugging these expressions into the original differential equation, we get:

y^{\prime \prime}_p-2 y^{\prime}_p-3 y_p=-3 t e^{-t}

0-2 (-A t e^{-t})-3 (A t e^{-t})=-3 t e^{-t}

2 A t e^{-t}-3 A t e^{-t}=-3 t e^{-t}

-A t e^{-t}=-3 t e^{-t}

This shows that $y_p=A t e^{-t}$ is indeed a solution of the non-homogeneous equation.

The General Solution

The general solution of the differential equation is the sum of the complementary solution and the particular solution:

y=y_c+y_p=c_1 e^{3 t}+c_2 e^{-t}+A t e^{-t}

Conclusion

In this article, we found the general solution of the differential equation $y^{\prime \prime}-2 y^{\prime}-3 y=-3 t e^{-t}$ . The general solution consists of two parts: the complementary solution and the particular solution. The complementary solution is the solution of the corresponding homogeneous equation, while the particular solution is a specific solution that satisfies the non-homogeneous equation. We verified that the particular solution satisfies the non-homogeneous equation and found the general solution by summing the complementary and particular solutions.

References

[1] Boyce, W. E., & DiPrima, R. C. (2017). Elementary differential equations and boundary value problems. John Wiley & Sons.
[2] Coddington, E. A., & Levinson, N. (1955). Theory of ordinary differential equations. McGraw-Hill.
[3] Simmons, G. F. (2011). Differential equations with applications and historical notes. McGraw-Hill.

Introduction

In our previous article, we found the general solution of the differential equation $y^{\prime \prime}-2 y^{\prime}-3 y=-3 t e^{-t}$ . In this article, we will answer some frequently asked questions related to finding the general solution of this differential equation.

Q: What is the difference between the complementary solution and the particular solution?

A: The complementary solution is the solution of the corresponding homogeneous equation, while the particular solution is a specific solution that satisfies the non-homogeneous equation.

Q: How do I find the complementary solution?

A: To find the complementary solution, you need to solve the corresponding homogeneous equation. In this case, the homogeneous equation is $y^{\prime \prime}-2 y^{\prime}-3 y=0$ . You can solve this equation using the characteristic equation, which is $r^2-2r-3=0$ . The roots of this equation are $r_1=3$ and $r_2=-1$ , so the complementary solution is of the form $y_c=c_1 e^{3 t}+c_2 e^{-t}$ .

Q: How do I find the particular solution?

A: To find the particular solution, you need to find a solution that satisfies the non-homogeneous equation. In this case, the non-homogeneous term is $-3 t e^{-t}$ . You can try to find a solution of the form $y_p=A t e^{-t}$ , where $A$ is a constant.

Q: How do I verify that the particular solution satisfies the non-homogeneous equation?

A: To verify that the particular solution satisfies the non-homogeneous equation, you need to plug it into the original differential equation and check if it satisfies the equation. In this case, we plugged in $y_p=A t e^{-t}$ and verified that it satisfies the equation.

Q: What is the general solution of the differential equation?

A: The general solution of the differential equation is the sum of the complementary solution and the particular solution. In this case, the general solution is $y=y_c+y_p=c_1 e^{3 t}+c_2 e^{-t}+A t e^{-t}$ .

Q: How do I use the general solution to solve a specific problem?

A: To use the general solution to solve a specific problem, you need to plug in the initial conditions and solve for the constants $c_1$ and $c_2$ . In this case, we assumed that the initial conditions were $y(0)=0$ and $y^{\prime}(0)=0$ . Plugging in these initial conditions, we found that $c_1=0$ and $c_2=0$ .

Q: What are some common mistakes to avoid when finding the general solution of a differential equation?

A: Some common mistakes to avoid when finding the general solution of a differential equation include:

Not checking if the particular solution satisfies the non-homogeneous equation
Not plugging in the initial conditions to solve for the constants
Not using the correct method to find the complementary solution
Not checking if the general solution satisfies the original differential equation

Q: What are some resources that can help me learn more about finding the general solution of a differential equation?

A: Some resources that can help you learn more about finding the general solution of a differential equation include:

Textbooks on differential equations, such as "Elementary Differential Equations and Boundary Value Problems" by William E. Boyce and Richard C. DiPrima
Online resources, such as Khan Academy and MIT OpenCourseWare
Practice problems and exercises, such as those found in "Differential Equations with Applications and Historical Notes" by George F. Simmons

Conclusion

In this article, we answered some frequently asked questions related to finding the general solution of the differential equation $y^{\prime \prime}-2 y^{\prime}-3 y=-3 t e^{-t}$ . We hope that this article has been helpful in clarifying some of the concepts and methods involved in finding the general solution of a differential equation.