Question Find the value of $y_0(2)$ where $y_0$ is the solution to the ODE $xy' + 2y = 2x$ with $y(1) = 3$ .

Studdy Solution

STEP 1

Assumptions
1. The given differential equation is $xy' + 2y = 2x$ .
2. The initial condition is $y(1) = 3$ .
3. We need to find the value of $y_0(2)$ , where $y_0$ is the solution to the differential equation satisfying the initial condition.

STEP 2

To solve the differential equation, we first need to identify its type. The given equation is a first-order linear differential equation in the form $xy' + p(x)y = q(x)$ , where $p(x) = 2/x$ and $q(x) = 2$ .

STEP 3

The integrating factor, $\mu(x)$ , for a first-order linear differential equation is given by
$\mu(x) = e^{\int p(x)dx}$

STEP 4

Calculate the integrating factor using the function $p(x) = 2/x$ .
$\mu(x) = e^{\int \frac{2}{x}dx}$

STEP 5

Integrate $2/x$ to find the integrating factor.
$\mu(x) = e^{2\ln|x|}$

STEP 6

Simplify the integrating factor using the properties of logarithms and exponentials.
$\mu(x) = e^{\ln|x|^2}$
$\mu(x) = |x|^2$
Since $x$ is in the domain where $x > 0$ (because we are given the initial condition at $x = 1$ ), we can drop the absolute value.
$\mu(x) = x^2$

STEP 7

Multiply the entire differential equation by the integrating factor $x^2$ .
$x^2(xy' + 2y) = x^2(2x)$

STEP 8

Simplify the equation.
$x^3y' + 2x^2y = 2x^3$

STEP 9

Recognize that the left-hand side of the equation is the derivative of a product of functions.
$\frac{d}{dx}(x^2y) = 2x^3$

STEP 10

Integrate both sides of the equation with respect to $x$ .
$\int \frac{d}{dx}(x^2y)dx = \int 2x^3dx$

STEP 11

Perform the integration.
$x^2y = \frac{2}{4}x^4 + C$
$x^2y = \frac{1}{2}x^4 + C$

STEP 12

Solve for $y$ by dividing both sides by $x^2$ .
$y = \frac{1}{2}x^2 + \frac{C}{x^2}$

STEP 13

Apply the initial condition $y(1) = 3$ to find the constant $C$ .
$3 = \frac{1}{2}(1)^2 + \frac{C}{(1)^2}$

STEP 14

Solve for $C$ .
$3 = \frac{1}{2} + C$
$C = 3 - \frac{1}{2}$
$C = \frac{5}{2}$

STEP 15

Substitute the value of $C$ back into the equation for $y$ .
$y = \frac{1}{2}x^2 + \frac{5/2}{x^2}$

STEP 16

Now that we have the general solution of the differential equation, we can find the value of $y_0(2)$ by plugging in $x = 2$ .
$y_0(2) = \frac{1}{2}(2)^2 + \frac{5/2}{(2)^2}$

STEP 17

Calculate the value of $y_0(2)$ .
$y_0(2) = \frac{1}{2}(4) + \frac{5/2}{4}$
$y_0(2) = 2 + \frac{5}{8}$
$y_0(2) = \frac{16}{8} + \frac{5}{8}$
$y_0(2) = \frac{21}{8}$
The value of $y_0(2)$ is $\frac{21}{8}$ .

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms