dy/dx=3y-7x+7/3x-7y-3 solve the following differential equations

Hello sakuntalaampolu,

Well, before I start solving it, I must admit it is going to be very lengthy as I am using bernoulli's homogeneous differential equation method:

Now, In first, if an equation is of form dy/dx = (ax + by + c)/(-bx+ay+f), then we should cross multiply and solve easily, But this is not the case here, So, put y = v + k, where v is variable and k is a constant, And put x = u + h, where u is variable and h is constant, and we get:

dy/dx = (3v-7u +3k-7h+7)/(3u-7v +3h-7k-3), now we know y = v + k => dy = dv and similarly du = dx So, dy/dx = dv/du and Now, lets make 3k-7h+7=0 and 3h-7k-3=0 because we want homogeneous equation and also k and h are constants while we will decide by solving above 2 equations having h and k , h=10/7 and k=1; Now we have dv/du = (3v-7u)/(3u-7v) Now, how do we solve homogeneous equation? => by putting v = ut where t is a variable, so dv/du = t + u.dt/du(you can also do dv/dt , No problem!) and we now get :

t + u.dt/du =  (3t-7)/(3-7t) => u.dt/du = 7(t^2 -1)/3-7t and rearranging we get:

((3-7t)/(7(t^2 -1))dt = du/u , Now Integrate both side, and you are going to get like : (-3/7 ln(t+1) -2/7 ln(t^2 -1) = ln u;( I hope you can solve it like 3-7t = (3-3t) -4t and rest you can do...)

And to get your answer, you have to put first t=v/u and then v = y-1(v = y-k and k=1) and u = x-10/7 and this can done by anyone,

And I am expecting you get the solution, Now Summarising:

first make homogenised equation by (using y=v+k and x=u+h) and after you get homogenised equation, always do like upper variable = t. lower variable like y = tx or v = ut and you can do the reverse also like u=vt, this will also give correct answer) and then you get a easily differentiable function to integrate and after integrating you should put the actual values or variables going in backward direction...

Good Luck!

Hope this helps, and feel free to ask any further query...