Chapter 4: Q3RP (page 231)

Find a general solution to the given differential equation. $4 y'' - 4 y' + 10 y = 0$

Short Answer

Expert verified

The general solution to the given differential equation is;

$y = c_{1} e^{\frac{1}{2} t} \cos (\frac{3 t}{2}) + c_{2} e^{\frac{1}{2} t} \sin (\frac{3 t}{2})$

Step by step solution

The general solution if the roots are complex conjugate

If the roots of the auxiliary equation are $α \pm i β$ , then the general solution is given as $y (t) = e^{α t} (c_{1} c o s β t + c_{2} s i n β t) .$

Write the auxiliary equation of the given differential equation

The given differential equation is $4 y'' - 4 y' + 10 y = 0 .$

The auxiliary equation for the above equation $4 m^{2} - 4 m + 10 = 0 .$

Find the roots of the auxiliary equation

Solve the auxiliary equation,

$\begin{matrix} 4 m^{2} - 4 m + 10 = 0 \\ m = \frac{- (- 4) \pm \sqrt{16 - 160}}{2 (4)} \\ m = \frac{4 \pm \sqrt{- 144}}{8} \\ m = \frac{4 \pm 12 i}{8} \\ m = \frac{1 \pm 3 i}{2} \end{matrix}$

The roots of the auxiliary equation are $m_{1} = \frac{1 + 3 i}{2} & m_{2} = \frac{1 - 3 i}{2} .$

Thus, the general solution of the given equation is $y = c_{1} e^{\frac{1}{2} t} \cos (\frac{3 t}{2}) + c_{2} e^{\frac{1}{2} t} \sin (\frac{3 t}{2}) .$

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Find a general solution to the given differential equation. $4 y'' - 4 y' + 10 y = 0$

Short Answer

Step by step solution

The general solution if the roots are complex conjugate

Write the auxiliary equation of the given differential equation

Find the roots of the auxiliary equation

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Find a general solution to the given differential equation.4y''-4y'+10y=0

Short Answer

Step by step solution

The general solution if the roots are complex conjugate

Write the auxiliary equation of the given differential equation

Find the roots of the auxiliary equation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Decision Maths

Statistics

Geometry

Theoretical and Mathematical Physics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Find a general solution to the given differential equation. $4 y'' - 4 y' + 10 y = 0$