2.6 Equations reducible to exact form

1.1 Order and degree of a differential equation
1.2 Formation of a differential equation

2.1 Variable separable method
2.2 Equations reducible to variable separable form
2.3 Homogeneous differential equation
2.4 Equations reducible to homogeneous form
2.5 Exact differential equation
2.6 Equations reducible to exact form
2.7 Linear differential equation
2.8 Equations reducible to linear form

If a differential equation of the form $$M(x,y)dx + N(x,y)dy = 0$$ is not exact can be converted to an exact form by multiplying by a suitable function $u(x,y)$ which is not identically zero. This function $u(x,y)$ is called the integrating factor. In order to find the integrating factors, following results must be remembered:

(i) $xdy + ydx = d(xy)$

(ii) $\frac{{xdy - ydx}}{{{x^2}}} = d\left( {\frac{y}{x}} \right)$

(iii) $2(xdx + ydy) = d({x^2} + {y^2})$

(iv) $\frac{{xdy - ydx}}{{xy}} = d\left( {\ln \frac{y}{x}} \right)$

(v) $\frac{{xdy - ydx}}{{{x^2} + {y^2}}} = d\left( {{{\tan }^{ - 1}}\frac{y}{x}} \right)$

(vi) $\frac{{xdy + ydx}}{{xy}} = d\left( {\ln xy} \right)$

(vii) $\frac{{ydx - xdy}}{{{y^2}}} = d\left( {\frac{x}{y}} \right)$

(viii) $\frac{{xdy + ydx}}{{{x^2}{y^2}}} = d\left( { - \frac{1}{{xy}}} \right)$

Question 10. Solve $$\frac{{x + y\frac{{dy}}{{dx}}}}{{y - x\frac{{dy}}{{dx}}}} = {x^2} + 2{y^2} + \frac{{{y^4}}}{{{x^2}}}$$

Solution: The given differential equation can be written as $$\frac{{xdx + ydy}}{{{{\left( {{x^2} + {y^2}} \right)}^2}}} = \frac{{ydx - xdy}}{{{y^2}}}.\frac{{{y^2}}}{{{x^2}}}$$

$$ \Rightarrow \int {\frac{{d({x^2} + {y^2})}}{{{{({x^2} + {y^2})}^2}}} = 2\int {\frac{1}{{\frac{{{x^2}}}{{{y^2}}}}}d\left( {\frac{x}{y}} \right)} } $$ Integrating both sides, we get $$ - \frac{1}{{({x^2} + {y^2})}} = - \frac{1}{{\left( {\frac{x}{y}} \right)}} + C $$ $$ \Rightarrow \frac{y}{x} - \frac{1}{{({x^2} + {y^2})}} = C $$