Chapter 4: Q34E (page 186)

Determine the form of a particular solution for the differential equation. Do not solve
$y'' + 5 y' + 6 y = s i n t - c o s 2 t$

Short Answer

Expert verified

Thus, the answer is: $y_{p} = A s i n t + B c o s t + C c o s 2 t + D s i n 2 t$

Step by step solution

Firstly, write the auxiliary equation of the given differential equation

The differential equation is,

$y'' + 5 y' + 6 y = s i n t - c o s 2 t . . . (1)$

Write the homogeneous differential equation of equation (1),

$y'' + 5 y' + 6 y = 0$

The auxiliary equation for the above equation,

$m^{2} + 5 m + 6 = 0$

Now find the complementary solution of the given equation is,

Solve the auxiliary equation,

$m^{2} + 5 m + 6 = 0 m^{2} + 3 m + 2 m + 6 = 0 m (m + 3) + 2 (m + 3) = 0 (m + 2) (m + 3) = 0$

The roots of the auxiliary equation are,

$m_{1} = - 2, m_{2} = - 3$

The complementary solution of the given equation is,

$y_{c} = c_{1} e^{- 2 t} + c_{2} e^{- 3 t}$

Now find the form of a particular solution

The particular solution to equation (1) will be the linear combination of terms $s i n t, c o s 2 t$ and their independent derivatives.

Therefore, the particular solution of equation (1),

$y_{p} = A s i n t + B c o s t + C c o s 2 t + D s i n 2 t$

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Determine the form of a particular solution for the differential equation. Do not solve
$y'' + 5 y' + 6 y = s i n t - c o s 2 t$

Short Answer

Step by step solution

Firstly, write the auxiliary equation of the given differential equation

Now find the complementary solution of the given equation is,

Now find the form of a particular solution

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Determine the form of a particular solution for the differential equation. Do not solvey''+5y'+6y=sint-cos2t

Short Answer

Step by step solution

Firstly, write the auxiliary equation of the given differential equation

Now find the complementary solution of the given equation is,

Now find the form of a particular solution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Discrete Mathematics

Pure Maths

Applied Mathematics

Statistics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Determine the form of a particular solution for the differential equation. Do not solve
$y'' + 5 y' + 6 y = s i n t - c o s 2 t$