# Matematik sorusunu çöz

**URL:** <https://sorumatik.co/t/matematik-sorusunu-coz/81285>\
**Category:** ÖABT\
**Tags:** matematik\
**Created:** [Haziran 25, 2024, 8:22öö UTC](https://sorumatik.co/t/matematik-sorusunu-coz/81285 "2024-06-25T08:22:00Z")\
**Posts on this page:** 2\
**Page:** 1

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**Author:** ![İbrahim\_Karakaş](https://sorumatik.co/user_avatar/sorumatik.co/i%CC%87brahim_karaka%C5%9F/32/27301_2.png) [@İbrahim\_Karakaş](https://sorumatik.co/u/%C4%B0brahim_Karaka%C5%9F)\
**Post date:** [Haziran 25, 2024, 8:22öö UTC](https://sorumatik.co/t/matematik-sorusunu-coz/81285/1 "2024-06-25T08:22:00Z")

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**Author:** ![Sorumatik\_Ogretmen](https://sorumatik.co/user_avatar/sorumatik.co/sorumatik_ogretmen/32/30517_2.png) [@Sorumatik\_Ogretmen](https://sorumatik.co/u/Sorumatik_Ogretmen)\
**Post date:** [Haziran 25, 2024, 8:22öö UTC](https://sorumatik.co/t/matematik-sorusunu-coz/81285/2 "2024-06-25T08:22:20Z")

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## **Solve the differential equation \frac{dy}{dx} + 2xy = 0**

**Answer:** The differential equation given is:

\frac{dy}{dx} + 2xy = 0

This is a first-order linear differential equation. To solve this, we can use the method of integrating factors.

### **Step 1: Identify the Integrating Factor**

The general form of a first-order linear differential equation is:

\frac{dy}{dx} + P(x)y = Q(x)

Here, P(x) = 2x and Q(x) = 0 .

The integrating factor \mu(x) is given by:

\mu(x) = e^{\int P(x) \, dx} = e^{\int 2x \, dx} = e^{x^2}

### **Step 2: Multiply the Differential Equation by the Integrating Factor**

Multiply both sides of the differential equation by e^{x^2} :

e^{x^2} \frac{dy}{dx} + 2x e^{x^2} y = 0

### **Step 3: Simplify and Integrate**

Notice that the left side of the equation is the result of the product rule:

\frac{d}{dx} \left( y e^{x^2} \right) = 0

Integrate both sides with respect to x :

\int \frac{d}{dx} \left( y e^{x^2} \right) \, dx = \int 0 \, dx

This simplifies to:

y e^{x^2} = C

where C is the constant of integration.

### **Step 4: Solve for y**

Finally, solve for y :

y = C e^{-x^2}

So, the general solution to the differential equation \frac{dy}{dx} + 2xy = 0 is:

y = C e^{-x^2}