# find general solution in semi homogenous method of dy/dx=x-y+1/x+y-1

Question
Differential equations
find general solution in semi homogenous method of $$\displaystyle\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}={x}-{y}+\frac{{1}}{{x}}+{y}-{1}$$

2021-01-17
The Given problem is $$\displaystyle\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}=\frac{{{x}-{y}+{1}}}{{{x}+{y}-{1}}}$$
Let us substitute X=x and Y=y-1 then dX=dx and dY=dY
Hence the above Differential equations reduces to PSKdY/dX=(X-Y)/(X+Y) ->(X+Y)dY=(X-Y)dX ->XdY+YdY=XdX-YdY ->XdX+YdX+YdY-XdX=0 ->d(XY)+YdY-XdX=0 ->∫d(XY)+∫YdY-∫XdX=C ->XY+((Y^2/2))-((X^2)/2)=CZSK
Substituting the values of X and Y $$\displaystyle\to{x}{\left({y}-{1}\right)}+{\left(\frac{{1}}{{2}}\right)}{\left({\left({y}-{1}\right)}^{{2}}-{x}^{{2}}\right)}={c}$$
This is the required general solution.

### Relevant Questions

Find the differential dy for the given values of x and dx. $$y=\frac{e^x}{10},x=0,dx=0.1$$
Find the solution $$\displaystyle{\left({x}\frac{{\sin{{y}}}}{{x}}\right)}{\left.{d}{y}\right.}={\left({y}\frac{{\sin{{y}}}}{{x}}-{x}\right)}{\left.{d}{x}\right.}$$
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.
$$\displaystyle{\left[\lambda_{{{1}}}=-{1}\Rightarrow\le{f}{t}{\left\lbrace{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}{0}{3}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{r}{i}{g}{h}{t}\right\rbrace},\lambda_{{{2}}}={3}{i}\Rightarrow\le{f}{t}{\left\lbrace{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{2}-{i}{1}+{i}{7}{i}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{r}{i}{g}{h}{t}\right\rbrace},\lambda_{{3}}=-{3}{i}\Rightarrow\le{f}{t}{\left\lbrace{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{2}+{i}{1}-{i}-{7}{i}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{r}{i}{g}{h}{t}\right\rbrace}\right]}$$
Find the general solution to the equation $$\displaystyle{x}{\left(\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}\right)}+{3}{\left({y}+{x}^{2}\right)}=\frac{{ \sin{{x}}}}{{x}}$$
Use logarithmic differentiation to find
dy/dx y=x sqrt{x^2+48}ZSK
Deterrmine the first derivative $$\displaystyle{\left(\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}\right)}$$ :
$$\displaystyle{y}={2}{e}^{{2}}{x}+{I}{n}{x}^{{3}}-{2}{e}^{{x}}$$
Q. 2# $$(x+1)\frac{dy}{dx}=x(y^{2}+1)$$
Q. 2# $$\displaystyle{\left({x}+{1}\right)}{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={x}{\left({y}^{{{2}}}+{1}\right)}$$
Solve. $$\displaystyle\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}+\frac{{3}}{{x}}{y}={27}{y}^{{\frac{{1}}{{3}}}}{1}{n}{\left({x}\right)},{x}{>}{0}$$