Resourcesdown chevron
Testimonials
Plans
Sign in
Sign up
Resourcesright arrow
Testimonials
Plans
Bytelearn - cat image with glassesAI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Solve the D.E: (dy)/(dx)+yx=-x^(3)

Solve the D.E: dydx+yx=−x3 \frac{d y}{d x}+y x=-x^{3}

View step-by-step helpView step-by-step help
Homeright chevron icon
Math Problemsright chevron icon
Algebra 2right chevron icon
Find Coordinate on Unit Circleright chevron icon

Full solution

Q. Solve the D.E: dydx+yx=−x3 \frac{d y}{d x}+y x=-x^{3}
  1. Identify type of equation: Identify the type of differential equation.\newlineThis is a first-order linear differential equation.
  2. Rewrite in standard form: Rewrite the equation in standard form. dydx+yx=−x3\frac{dy}{dx} + yx = -x^3 can be written as dydx+xy=−x3\frac{dy}{dx} + xy = -x^3.
  3. Find integrating factor: Find the integrating factor, μ(x)\mu(x).μ(x)=e(∫x dx)=ex22\mu(x) = e^{(\int x \, dx)} = e^{\frac{x^2}{2}}.
  4. Multiply by integrating factor: Multiply the entire differential equation by the integrating factor.\newlineex22⋅dydx+ex22⋅xy=−x3⋅ex22e^{\frac{x^2}{2}} \cdot \frac{dy}{dx} + e^{\frac{x^2}{2}} \cdot xy = -x^3 \cdot e^{\frac{x^2}{2}}.
  5. Recognize derivative of product: Recognize the left side of the equation as the derivative of a product. ddx[y⋅e(x2/2)]=−x3⋅e(x2/2).\frac{d}{dx} [y \cdot e^{(x^2/2)}] = -x^3 \cdot e^{(x^2/2)}.
  6. Integrate both sides: Integrate both sides with respect to xx.∫ddx[y⋅e(x2/2)]dx=∫−x3⋅e(x2/2)dx.\int \frac{d}{dx} [y \cdot e^{(x^2/2)}] dx = \int -x^3 \cdot e^{(x^2/2)} dx. y⋅e(x2/2)=−∫x3⋅e(x2/2)dx+C.y \cdot e^{(x^2/2)} = -\int x^3 \cdot e^{(x^2/2)} dx + C.
  7. Solve integral: Solve the integral on the right side.\newlineThis step involves an error in integration; the integral of −x3⋅ex2/2-x^3 \cdot e^{x^2/2} is not elementary and cannot be expressed in terms of elementary functions.

More problems from Find Coordinate on Unit Circle

Question
Find all solutions with 0∘≤θ≤180∘0^\circ \leq \theta \leq 180^\circ. Give the exact answer(s) in simplest form. If there are multiple answers, separate them with commas.\newlinecos⁡(θ)=−1\cos(\theta) = -1\newline‾∘\underline{\hspace{2em}}^\circ
Get tutor helpright-arrow

Posted 1 month ago

Question
Kimi's school is 1515 kilometers west of her house and 88 kilometers south of her friend Franklin's house. Every day, Kimi bicycles from her house to her school. After school, she bicycles from her school to Franklin's house. Before dinner, she bicycles home on a bike path that goes straight from Franklin's house to her own house. How far does Kimi bicycle each day?\newline_____\_\_\_\_\_ kilometers
Get tutor helpright-arrow

Posted 1 month ago

Question
Evaluate. Write your answer as an integer or as a decimal rounded to the nearest hundredth.\newlinecos⁡35∘=\cos 35^\circ = __
Get tutor helpright-arrow

Posted 1 month ago

Question
Find all solutions with −90∘≤θ≤90∘-90^\circ \leq \theta \leq 90^\circ. Give the exact answer(s) in simplest form. If there are multiple answers, separate them with commas.\newlinecsc⁡(θ)=−1\csc(\theta) = -1\newline____ ∘\_\_\_\_\,^\circ
Get tutor helpright-arrow

Posted 1 month ago

Question
Evaluate. Write your answer in simplified, rationalized form. Do not round.\newlinecot⁡30∘=\cot 30^\circ = ______
Get tutor helpright-arrow

Posted 1 month ago

Question
The terminal side of an angle θ\theta in standard position intersects the unit circle at (8485,1385)(\frac{84}{85}, \frac{13}{85}). What is cos⁡(θ)\cos(\theta)?\newlineWrite your answer in simplified, rationalized form.\newline______
Get tutor helpright-arrow

Posted 1 month ago

Question
−90∘<θ<90∘-90^\circ < \theta < 90^\circ. Find the value of θ\theta in degrees.\newlinetan⁡(θ)=0\tan(\theta) = 0\newlineWrite your answer in simplified, rationalized form. Do not round.\newlineθ=\theta = ____∘^\circ\newline
Get tutor helpright-arrow

Posted 2 months ago

Question
Evaluate. Write your answer in simplified, rationalized form. Do not round.\newlinecsc⁡60∘=\csc 60^\circ = ______
Get tutor helpright-arrow

Posted 1 month ago

Question
θ\theta is an acute angle. Find the value of θ\theta in degrees.\newlinesec⁡(θ)=2\sec(\theta) = \sqrt{2}\newlineθ=\theta = ____°\degree
Get tutor helpright-arrow

Posted 1 month ago

Question
Evaluate. Write your answer in simplified, rationalized form. Do not round.\newlinecos⁡90∘=\cos 90^\circ = ____
Get tutor helpright-arrow

Posted 2 months ago

View all (68)