Answer in Differential Equations for Mubina #139428

“displaystyletextsf{A first order differential equation is said}\textsf{to be homogeneous if it may be written as}\nn{displaystyle f(x,y)mathrm{d}y=g(x,y)mathrm{d}x,}\nnntextsf{where}, f , textsf{and}, g , textsf{are homogeneous functions}\textsf{of the same degree of}, x , textsf{and}, y.\nnnn textsf{The functions are homogeneous functions of the}\textsf{same degree of}, x, textsf{and}, y \ textsf{if they satisfies the condition} \nn{displaystyle f(alpha x,alpha y)=alpha ^{k}f(x,y)}\nn{displaystyle g(beta x,beta y)=beta^{k}g(x,y)}\ntextsf{For some constant}, k , textsf{and all real numbers}, alpha, beta.\ textsf{The constant}, k, textsf{is called the degree of homogeneity.}\nnntextbf{textsf{Example}}\nnfrac{mathrm{d}y}{mathrm{d}x} + 4xy = 0\nnsin(x)frac{mathrm{d}^2y}{mathrm{d}x^2} + 6frac{mathrm{d}y}{mathrm{d}x} + y = 0\nnntextsf{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.}\nntextsf{The nonhomogeneous differential equations}\textsf{is in this format:}\nnyu201d + p(x)y’ + q(x)y = g(x).\nnntextbf{textsf{Examples}}\nnnnfrac{mathrm{d}^2y}{mathrm{d}x^2} – 4frac{mathrm{d}y}{mathrm{d}x} + 5y = cos(x)\nnnfrac{mathrm{d}^2y}{mathrm{d}x^2} – n^2y = 2 + sin(7x)\nnntextsf{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.}\nnntextsf{If this equation can be solved for},y’\ntextsf{we get one or several explicit}\textsf{differential equations of type}\ny’ = f(x, y)\nntextbf{textsf{Examples}}\nn9(y’)^2 – 4x = 0\nnny = ln(25 + (y’)^2)”

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