Solve this equation pls y'+xy=e^x y(0)=1

Solve this equation pls y'+xy=e^x y(0)=1

Question
Differential equations
asked 2021-01-02
Solve this equation pls \(\displaystyle{y}'+{x}{y}={e}^{{x}}\)
\(\displaystyle{y}{\left({0}\right)}={1}\)

Answers (1)

2021-01-03
First, let us find the solution of the homogeneous equation
\(\displaystyle{y}'+{x}{y}={0}\)
We get,
\(\displaystyle{y}'+{x}{y}={0}\)
\(\displaystyle\Rightarrow{y}'=-{x}{y}\)
\(\displaystyle\Rightarrow\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}=-{x}{y}\)
\(\displaystyle\Rightarrow\frac{{\left.{d}{y}\right.}}{{y}}={\left(-{x}\right)}{\left.{d}{x}\right.}\)
By integrating both sides, we get,
\(\displaystyle\int\frac{{\left.{d}{y}\right.}}{{y}}=\int{\left(-{x}\right)}{\left.{d}{x}\right.}\)
\(\displaystyle\Rightarrow{\ln{{\left({y}\right)}}}-\frac{{x}^{{2}}}{{2}}+{\ln{{\left({C}\right)}}}\)
\(\displaystyle\Rightarrow{\log{{\left({y}\right)}}}-{\log{{\left({C}\right)}}}=-\frac{{x}^{{2}}}{{2}}\)
\(\displaystyle\Rightarrow{\left(\frac{{y}}{{C}}\right)}=-\frac{{x}^{{2}}}{{2}}\)
\(\displaystyle\Rightarrow{y}={C}{e}^{{\frac{{x}^{{2}}}{{2}}}}\)
So, the general solution of the given differential equation is given by
\(\displaystyle{a}{e}^{{-\frac{{x}^{{2}}}{{2}}}}+{e}^{{-\frac{{x}^{{2}}}{{2}}}}\int\frac{{{e}^{{x}}}}{{{e}^{{-\frac{{x}^{{2}}}{{2}}}}}}{\left.{d}{x}\right.}\)
image
image
image
0

Relevant Questions

asked 2021-05-10
Solve the equation:
\(\displaystyle{\left({a}-{x}\right)}{\left.{d}{y}\right.}+{\left({a}+{y}\right)}{\left.{d}{x}\right.}={0}\)
asked 2021-03-22
Solve the equation:
\(\displaystyle{\left({x}+{1}\right)}{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={x}{\left({y}^{{2}}+{1}\right)}\)
asked 2021-05-01
Solve differential equation \(xy'+2y= -x^3+x, \ y(1)=2\)
asked 2021-02-05
Solve the differential equations
(1) \(\displaystyle{x}{y}'-{2}{y}={x}^{{3}}{e}^{{x}}\)
(2) \(\displaystyle{\left({2}{y}{\left.{d}{x}\right.}+{\left.{d}{y}\right.}\right)}{e}^{{2}}{x}={0}\)
asked 2021-02-25
Give the correct answer and solve the given equation: \(\displaystyle{y}\ \text{ - 4y}+{3}{y}={x},{y}_{{1}}={e}^{x}\)
asked 2021-05-25
Random variables X and Y are uniformly distributed on the region bounded by the x and y axes, and the curve \(y=1-x^{2}\).
Calculate E(XY)
asked 2021-02-21
Solve the given system of differential equations.
\[Dx+Dy+(D+1)z=0\)
Dx+y=e^{t}\)
Dx+y-2z=50\sin(2t)\)
asked 2021-01-02
Solve \(y''\ +\ 3y'\ -\ 10y=x(e^{x}\ +\ 1)\)
asked 2020-12-05
Give the correct answer and solve the given equation:
\(\displaystyle{x}{y}{\left.{d}{x}\right.}-{\left({y}+{2}\right)}{\left.{d}{y}\right.}={0}\)
asked 2021-05-12
Replace the Cartesian equation with equivalent polar equations.
\(x^{2}+xy+y^{2}=1\)
...