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 . $erf (x) = \frac{2}{\sqrt{π}} \int_{0}^{x} e^{- t^{2}} d t$

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:

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

Now integrating both sides as:

$\begin{matrix} \int \frac{d p}{p} = - 2 \int x d x \Rightarrow p = e^{- x^{2} + c} \\ \ln p = - 2 \cdot \frac{x^{2}}{2} + c \Rightarrow p = e^{- x^{2}} \cdot e^{c} \\ \ln p = - x^{2} + c \Rightarrow p = A e^{- x^{2}} \end{matrix}$

Where . $A = e^{c}$

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

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

Integrating both sides as:

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

Now using. $\int e^{- x^{2}} d x = \frac{1}{2} \sqrt{π} erf (x)$

So, integrating as:

$\begin{matrix} \int d y = A \int e^{- x^{2}} d x \\ y = A \times \frac{1}{2} \sqrt{π} erf (x) + c_{2} \\ y = c_{1} erf (x) + c_{2} \end{matrix}$

Where. $c_{1} = \frac{A}{2} \sqrt{π}$

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

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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.

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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

Dynamics

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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 $erf x$ .