Differential Equations
Homogeneous and Linear Differential Equations
- A differential equation which can be expressed in the form \frac{dy}{dx}=f(x,y)\ or\ \frac{dx}{dy}=g(x,y) where, f(x,y) and g(x,y) are homogenous functions of degree zero is called a homogeneous differential equation.
- If the homogeneous differential equation is in the form \frac{dx}{dy}= F(x, y) where, F(x, y) is homogeneous function of degree zero, then we make substitution \frac{x}{y}=v i.e., x = vy and we proceed further to find the general solution as discussed above by writing \frac{dx}{dy}=F(x,y)=h \left(\frac{x}{y}\right).
- A differential equation of the form \frac{dy}{dx}+Py=Q, where P and Q are constants or functions of x only is called a first order linear differential equation.
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- \frac{dy}{dx} + P(x) y = Q(x) ⇒ If = e^{\int P(x) \ dx}
- \frac{dx}{dy} + P(y) x = Q(y) ⇒ If = e^{\int P(y) \ dy}
- d (x + y) = dx + dy
- d(xy) = y dx + x dy
- d\left(\frac{x}{y}\right)=\frac{y dx- x dy}{x^{2}}
- d (x2 + y2) = 2 (x dx + y dy)
- d\left(\frac{y}{x}\right)=\frac{x dy- y dx}{x^{2}}
- d\left(\log(xy)\right)=\frac{y dx+ x dy}{xy}
- d\left(\log\left(\frac{y}{x}\right)\right)=\frac{x dy - y dx}{xy}=\frac{-dx}{x}+\frac{dy}{y}
- d\left(\log\left(\frac{x}{y}\right)\right)=\frac{y dx - x dy}{xy}=\frac{dx}{x}-\frac{dy}{y}
- d\left(\tan^{-1}\left(\frac{y}{x}\right)\right)=\frac{x dy - y dx}{x^{2}+y^{2}}=\frac{\left(\frac{x dy - y dx}{x^{2}}\right)}{\left(1+\frac{y^{2}}{x^{2}}\right)}=\frac{d\left(\frac{y}{x}\right)}{1+\left(\frac{y}{x}\right)^2}
- d\left(\log(x^{2}+y^{2})\right)=2\left(\frac{x dx + y dy}{x^{2}+y^{2}}\right)
- d\left(\sqrt{x^{2}+y^{2}}\right)=\frac{x dx + y dy}{\sqrt{x^{2}+y^{2}}}
- d\left(\frac{-1}{xy}\right)=\frac{x dy + y dx}{x^{2}y^{2}}