# Solve differential equation dx/dy+x/y= 1/(sqrt(1+y^2))

Question
Solve differential equation $$dx/dy+x/y= 1/(sqrt(1+y^2))$$

2020-11-28
$$dx/dy+p(y)x=q(y)$$
Compare the equation $$dx/dy+p(y)x=q(y)$$ to $$dx/dy+x/y= 1/(sqrt(1+y^2))$$ and obtain $$p(y)=1/y$$, $$q(y)= 1/(sqrt(1+y^2))$$
$$I.F.= e^(int p(y)dy)$$
$$= e^(ln y)$$
=y $$xe^(int p(y)dy)= int e^(int p(y)dy)q(y)dy+C$$
where C is arbitrary constant of equation
$$xe^(int p(y)dy)= int e^(int p(y)dy)q(y)dy+C$$
$$xy= int y 1/(sqrt(1+y^2))dy+C$$
$$xy= 1/2 int (2y)/(sqrt(1+y^2)) dy+C$$
$$xy= 1/2 int ((1+y^2)')/(sqrt(1+y^2)) dy+C$$ [$$:' (1+y^2)'= 2y$$]
$$xy= 1/2 (2 sqrt(1+y^2))+C$$ [$$:' int (f(y))/(sqrt(f(y)))dy= 2 sqrt(f(y))$$]
$$xy= sqrt(1+y^2)+C$$
$$x= (sqrt(1+y^2)+C)/y$$
$$x= (sqrt(1+y^2))/y+C/y$$

### Relevant Questions

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

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

Consider the curves in the first quadrant that have equationsy=Aexp(7x), where A is a positive constant. Different valuesof A give different curves. The curves form a family,F. Let P=(6,6). Let C be the number of the family Fthat goes through P.
A. Let y=f(x) be the equation of C. Find f(x).
B. Find the slope at P of the tangent to C.
C. A curve D is a perpendicular to C at P. What is the slope of thetangent to D at the point P?
D. Give a formula g(y) for the slope at (x,y) of the member of Fthat goes through (x,y). The formula should not involve A orx.
E. A curve which at each of its points is perpendicular to themember of the family F that goes through that point is called anorthogonal trajectory of F. Each orthogonal trajectory to Fsatisfies the differential equation dy/dx = -1/g(y), where g(y) isthe answer to part D.
Find a function of h(y) such that x=h(y) is the equation of theorthogonal trajectory to F that passes through the point P.
Solve differential equation $$\displaystyle{y}{\left({2}{x}-{2}+{x}{y}+{1}\right)}{\left.{d}{x}\right.}+{\left({x}-{y}\right)}{\left.{d}{y}\right.}={0}$$
Solve differential equation $$\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={\left({x}+{y}+{1}\right)}^{{2}}-{\left({x}+{y}-{1}\right)}^{{2}}$$
Solve differential equation $$\displaystyle{2}{x}{y}-{9}{x}^{{2}}+{\left({2}{y}+{x}^{{2}}+{1}\right)}{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={0},\ {y}{\left({0}\right)}=-{3}$$
Solve differential equation $$2xy-9x^2+(2y+x^2+1)dy/dx=0$$, y(0)= -3
Solve differential equation $$dy/dx= (y^2-1)/(x^2-1)$$
Solve differential equation $$1+y^2+xy^2)dx+(x^2y+y+2xy)dy=0$$
Solve differential equation $$(y2+1)dx=ysec^2(x)dy$$, y(0)=0