Solve differential equationy'+y/x= 3y^(-2) x>0

Question

Solve differential equation $$y'+y/x= 3y^{-2}$$ $$x>0$$

2020-12-14

Dividing both sidesby $$y^{-2}$$
$$y^2y'+ y^3/x=3$$
$$u=y^3$$
Differentiating both sides with respect to x
$$du/dx= 3y^2 dy/dx$$
$$1/3 (du)/dx= y^2 dy/dx$$
Substituting this in the equation $$y^2y'+ y^3/x=3$$
Substituting $$1/3 (du)/dx= y^2 dy/dx$$ and $$u=y^3$$ in the equation $$y^2y'+ y^3/x=3$$
$$1/3 (du)/dx+ u/x=3$$
$$1/9 (du)/dx+u/(3x)=1$$

Relevant Questions

Solve differential equation $$\frac{\cos^2y}{4x+2}dy= \frac{(\cos y+\sin y)^2}{\sqrt{x^2+x+3}}dx$$

Write the first order differential equation for $$y=2-\int_0^x(1+y(t))\sin tdt$$

Write an equivalent first-order differential equationand initial condition for y $$y= 1+\int_0^x y(t) dt$$

Solve differential equation $$dy+5ydx=e^{-5x}dx$$

The coefficient matrix for a system of linear differential equations of the form $$\displaystyle{y}^{{{1}}}={A}_{{{y}}}$$ has the given eigenvalues and eigenspace bases. Find the general solution for the system.
$$\left[\lambda_{1}=-1\Rightarrow\left\{\begin{bmatrix}1 0 3 \end{bmatrix}\right\},\lambda_{2}=3i\Rightarrow\left\{\begin{bmatrix}2-i 1+i 7i \end{bmatrix}\right\},\lambda_3=-3i\Rightarrow\left\{\begin{bmatrix}2+i 1-i -7i \end{bmatrix}\right\}\right]$$

Determine whether the ordered pair is a solution to the given system of linear equations.
(1,2)
$$\left\{\begin{matrix} 3x−y=1 \\ 2x+3y=8 \end{matrix}\right\}$$

What is the process to solve these:
Vertex, $$y-\int$$., $$x-\int$$, graph $$= -(x+1)^2+1$$

Find the solution (x,y)to the system of equations.
$$\begin{cases}3x + y = 28\\-x+3y=4\end{cases}$$
Multiply the coordinates $$x \cdot y$$

Let $$\displaystyle{f{{\left({x},{y}\right)}}}=-\frac{{{x}{y}}}{{{x}^{{2}}+{y}^{{2}}}}$$.
Find limit of $$f(x,y)\ \text{as}\ (x,y)\ \rightarrow (0,0)\ \text{i)Along y axis and ii)along the line}\ y=x.\ \text{Evaluate Limes}\ \lim_{x,y\rightarrow(0,0)}y\log(x^{2}+y^{2})$$,by converting to polar coordinates.
$$2x+3y=-4$$
$$3x+2y=-1$$