Question

Explain homogeneous and non-homogeneous differential equation and implicit differential equation each with two examples?

Answer 1

$\displaystyle\textsf{A first order differential equation is said}\\\textsf{to be homogeneous if it may be written as}\\ {\displaystyle f(x,y)\mathrm{d}y=g(x,y)\mathrm{d}x,}\\ \textsf{where}\, f \, \textsf{and}\, g \, \textsf{are homogeneous functions}\\\textsf{of the same degree of}\, x \, \textsf{and}\, y.\\ \textsf{The functions are homogeneous functions of the}\\\textsf{same degree of}\, x\, \textsf{and}\, y \\ \textsf{if they satisfies the condition} \\ {\displaystyle f(\alpha x,\alpha y)=\alpha ^{k}f(x,y)}\\ {\displaystyle g(\beta x,\beta y)=\beta^{k}g(x,y)}\\ \textsf{For some constant}\, k \, \textsf{and all real numbers}\, \alpha, \beta.\\ \textsf{The constant}\, k\, \textsf{is called the degree of homogeneity.}\\ \textbf{\textsf{Example}}\\ \frac{\mathrm{d}y}{\mathrm{d}x} + 4xy = 0\\ \sin(x)\frac{\mathrm{d}^2y}{\mathrm{d}x^2} + 6\frac{\mathrm{d}y}{\mathrm{d}x} + y = 0\\ \textsf{Nonhomogeneous differential equations are}\\\textsf{the same as homogeneous differential equations,}\\\textsf{except they can have terms involving only},x \\\textsf{(and constants) on the right side,}\\\textsf{as in this equation.}\\ \textsf{The nonhomogeneous differential equations}\\\textsf{is in this format:}\\ y” + p(x)y' + q(x)y = g(x).\\ \textbf{\textsf{Examples}}\\ \frac{\mathrm{d}^2y}{\mathrm{d}x^2} - 4\frac{\mathrm{d}y}{\mathrm{d}x} + 5y = \cos(x)\\ \frac{\mathrm{d}^2y}{\mathrm{d}x^2} - n^2y = 2 + \sin(7x)\\ \textsf{An equation of type}\, F(x, y, y') \, \textsf{where}\, F \, \textsf{is a}\\\textsf{continuous function, is called the}\\\textsf{first order implicit differential equation.}\\ \textsf{If this equation can be solved for}\,y'\\ \textsf{we get one or several explicit}\\\textsf{differential equations of type}\\ y' = f(x, y)\\ \textbf{\textsf{Examples}}\\ 9(y')^2 - 4x = 0\\ y = \ln(25 + (y')^2)$