Q2P Solve the following differential... [FREE SOLUTION]

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Chapter 8: Q2P (page 423)

Solve the following differential equations by method (a) or (b) above. $y^{''} + 2 x y^{'} = 0$
. Hint: The solution is $y = c_{1} erf x + c_{2}$ see Chapter 11, Section 9 for the definition of $erf x$ .

Short Answer

Expert verified

The general solution of $y^{''} + 2 x y^{'} = 0$ is. $y = c_{1} erf (x) + c_{2}$

Step by step solution

Given Information.

The given equation is $y^{''} + 2 x y^{'} = 0$ .

Definition/ Concept.

The formula for . $erf (x) = \frac{2}{\sqrt{蟺}} {鈭�}_{0}^{x} e^{鈭� t^{2}} d t$

Solve the differential equation.

Using the method (a).

Let $y^{'} = p$ and . $y^{''} = p^{'}$

Now put these values in the given equation as:

$\begin{matrix} y^{''} + 2 x y^{'} = 0 \\ p^{'} + 2 x p = 0 \\ \frac{d p}{d x} + 2 x p = 0 \end{matrix}$

Now separating variables as:

$\begin{matrix} \frac{d p}{d x} + 2 x p = 0 \\ \frac{d p}{d x} = 鈭� 2 x p \\ \frac{d p}{p} = 鈭� 2 x d x \end{matrix}$

Now integrating both sides as:

$\begin{matrix} 鈭� \frac{d p}{p} = 鈭� 2 鈭� x d x 鈬� p = e^{鈭� x^{2} + c} \\ \ln p = 鈭� 2 鈰� \frac{x^{2}}{2} + c 鈬� p = e^{鈭� x^{2}} 鈰� e^{c} \\ \ln p = 鈭� x^{2} + c 鈬� p = A e^{鈭� x^{2}} \end{matrix}$

Where . $A = e^{c}$

Now put $p = y^{'}$ again $p = c_{1} e^{鈭� x^{2}}$ in as:

$\begin{matrix} p = c_{1} e^{鈭� x^{2}} \\ y^{'} = A e^{鈭� x^{2}} \\ \frac{d y}{d x} = A e^{鈭� x^{2}} \\ d y = A e^{鈭� x^{2}} d x \end{matrix}$

Integrating both sides as:

$\begin{matrix} d y = c_{1} e^{鈭� x^{2}} d x \\ 鈭� d y = A 鈭� e^{鈭� x^{2}} d x \end{matrix}$

Now using. $鈭� e^{鈭� x^{2}} d x = \frac{1}{2} \sqrt{蟺} erf (x)$

So, integrating as:

$\begin{matrix} 鈭� d y = A 鈭� e^{鈭� x^{2}} d x \\ y = A 脳 \frac{1}{2} \sqrt{蟺} erf (x) + c_{2} \\ y = c_{1} erf (x) + c_{2} \end{matrix}$

Where. $c_{1} = \frac{A}{2} \sqrt{蟺}$

Hence, the general solution of $y^{''} + 2 x y^{'} = 0$ is. $y = c_{1} erf (x) + c_{2}$

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91影视

Solve the following differential equations by method (a) or (b) above. $y^{''} + 2 x y^{'} = 0$
. Hint: The solution is $y = c_{1} erf x + c_{2}$ see Chapter 11, Section 9 for the definition of $erf x$ .

Short Answer

Step by step solution

Given Information.

Definition/ Concept.

Solve the differential equation.

One App. One Place for Learning.

Most popular questions from this chapter

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91影视

Solve the following differential equations by method (a) or (b) above.y''+2xy'=0. Hint: The solution isy=c1erfx+c2see Chapter 11, Section 9 for the definition of erfx.

Short Answer

Step by step solution

Given Information.

Definition/ Concept.

Solve the differential equation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Electricity and Magnetism

Electrostatics

Thermodynamics

Linear Momentum

Conservation of Energy and Momentum

Radiation

Study anywhere. Anytime. Across all devices.

Solve the following differential equations by method (a) or (b) above. $y^{''} + 2 x y^{'} = 0$
. Hint: The solution is $y = c_{1} erf x + c_{2}$ see Chapter 11, Section 9 for the definition of $erf x$ .