Chapter 4: Q8E (page 186)

A nonhomogeneous equation and a particular solution are given. Find a general solution for the equation.
$y'' = 2 y' - y + 2 e^{x}, y_{p} (x) = x^{2} e^{x}$

Short Answer

Expert verified

The general solution of the given differential equation is $y = c_{1} e^{x} + c_{2} x e^{x} + x^{2} e^{x}$

Step by step solution

Write the auxiliary equation of the given differential equation.

The differential equation is,

$y'' = 2 y' - y + 2 e^{x} y'' - 2 y' + y = 2 e^{x} \dots (1)$

Write the homogeneous differential equation of the equation (1),

$y'' - 2 y' + y = 0$

The auxiliary equation for the above equation,

$m^{2} - 2 m + 1 = 0$

Now find the complementary solution of the given equation.

Solve the auxiliary equation,

$m^{2} - 2 m + 1 = 0 {(m - 1)}^{2} = 0$

The roots of the auxiliary equation are,

$m_{1} = 1, m_{2} = 1$

The complementary solution of the given equation is,

$y_{c} = c_{1} e^{x} + c_{2} x e^{x}$

Use the given particular solution to find a general solution for the equation.

The given particular solution,

$y_{p} (x) = x^{2} e^{x}$

Therefore, the general solution is,

$y = y_{c} (x) + y_{p} (x) y = c_{1} e^{x} + c_{2} x e^{x} + x^{2} e^{x}$

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A nonhomogeneous equation and a particular solution are given. Find a general solution for the equation.
$y'' = 2 y' - y + 2 e^{x}, y_{p} (x) = x^{2} e^{x}$

Short Answer

Step by step solution

Write the auxiliary equation of the given differential equation.

Now find the complementary solution of the given equation.

Use the given particular solution to find a general solution for the equation.

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A nonhomogeneous equation and a particular solution are given. Find a general solution for the equation.y''=2y'-y+2ex, ypx=x2ex

Short Answer

Step by step solution

Write the auxiliary equation of the given differential equation.

Now find the complementary solution of the given equation.

Use the given particular solution to find a general solution for the equation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Theoretical and Mathematical Physics

Probability and Statistics

Calculus

Discrete Mathematics

Geometry

Study anywhere. Anytime. Across all devices.

A nonhomogeneous equation and a particular solution are given. Find a general solution for the equation.
$y'' = 2 y' - y + 2 e^{x}, y_{p} (x) = x^{2} e^{x}$