Solve differential equation xy'+2y= -x^3+x, \ y(1)=2

Lennie Carroll 2021-05-01 Answered
Solve differential equation \(xy'+2y= -x^3+x, \ y(1)=2\)

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Plainmath recommends

  • Ask your own question for free.
  • Get a detailed answer even on the hardest topics.
  • Ask an expert for a step-by-step guidance to learn to do it yourself.
Ask Question

Expert Answer

gotovub
Answered 2021-05-02 Author has 14930 answers

Divide both sides by x
\(y'+ \frac{2}{x}y= -x^2+1\)
This is first order linear differential equation of form
\(p(x)= \frac{2}{x}, \ q(x)= -x^2+1\)
Integrating factor is
\(=e^{\int p(x)dx}\)
\(=e^{\int \frac{2}{x}dx}\)
\(=e^{2 logx}\)
\(=e^{log x^2}\)
\(=x^2\)
equation is
\(y \cdot (I.F.)= \int (I.F.) q(x)dx+c\)
\(yx^2= \int x^2(-x+1)dx+c\)
\(yx^2=\int-x^4+x^2dx+c\)
\(x^2y= -\frac{x^5}{5}+\frac{x^3}{3}+c\)
\(y= -\frac{x^3}{5}+\frac{x}{3}+cx^{-2}\)
Given initial condition
\(y(1)=2\)
\(2= -\frac{1}{5}+\frac{1}{3}+\frac{c}{1}\)
\(2= \frac{-3+5}{15}+c\)
\(2= \frac{2}{15}+c\)
\(2- \frac{2}{15}=c\)
\(\frac{30-2}{15}=c\)
\(\frac{28}{15}=c\)
\(y= -\frac{x^3}{5}+\frac{x}{3}+\frac{28}{15x^2}\)

Have a similar question?
Ask An Expert
29
 

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Relevant Questions

asked 2021-05-26
Write the following first-order differential equations in standard form. \(\displaystyle{y}'={x}^{{{3}}}{y}+{\sin{{x}}}\)
asked 2021-05-03
How do you solve linear first-order differential equations?
asked 2021-05-22
Write the following first-order differential equations in standard form. \(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{t}\right.}}}}={y}{x}{\left({x}+{1}\right)}\)
asked 2021-07-05
Write first-order differential equations that express the situation: The increase in the number of people who have heard a rumor in a town of population 20,00020,000 is proportional to the product of the number PP who have heard the rumor and the number who have not heard the rumor at time tt .
asked 2021-06-10
Write first-order differential equations that express the situation: A student in an engineering course finds the rate of increase of his grade point average GG is directly proportional to the number NN of study hours/week and inversely proportional to the amount AA of time spent on online games .
asked 2021-03-07

Solve differential equation \((1+y^2+xy^2)dx+(x^2y+y+2xy)dy=0\)

asked 2021-08-31
Consider the Chebyshev differential equation
\(\displaystyle{\left({1}-{x}^{{2}}\right)}{y}{''}-{x}{y}'+\lambda^{{2}}{y}={0}\)
where \(\displaystyle\lambda\) is a constant. Find the power series solutions of this differential equation about \(\displaystyle{x}_{{0}}={0}\). For what values of the constant \(\displaystyle\lambda\) these solutions reduce to polynomials, known as Chebyshev polynomials. Compute first four of the Chebyshev polynomials.

Plainmath recommends

  • Ask your own question for free.
  • Get a detailed answer even on the hardest topics.
  • Ask an expert for a step-by-step guidance to learn to do it yourself.
Ask Question
...