An Equation Of The Form \(\frac{d y}{d x}=f(x, y)\) is called a differential equation of the first order and of the first degree.
We study the following four methods for solving \(\frac{d y}{d x}=f(x, y)\)
1. Variables separable.
2. Homogeneous equations and equations reducible to homogeneous form.
3. Exact equations can be made exact by the use of integrating factors.
4. Linear equations and Bemouli’s form.
Before discussing the methods for solving \(\frac{d y}{d x}=f(x, y)\) without proof concerning the existence and uniqueness of solutions.
Differential Equations Introduction Existence And Uniqueness Theorem
Let S denote the rectangular region defined by \(\left|x-x_0\right| \leq a\) and \(\left|y-y_0\right| \leq b\) a region with the point \(\left(x_0, y_0\right)\) at its centre. If f(x,y) and \(\frac{\partial f}{\partial y}\) are continuous functions of x and y in a region S of the xy – plane and if \(\mathrm{P}\left(x_0, y_0\right) \in\) S, then there exists one and only one function say Φ(x), which in some neighbourhood of P, is a solution of the differential equation \(\frac{d y}{d x}=f(x, y)\) and is such that \(\phi\left(x_0\right)=y_0\)
In other words, the theorem states that if f(x,y) and \(\frac{\partial f}{\partial y}\) are continuous functions of x and y in a region S., then the differential equation \(\frac{d y}{d x}=f(x, y)\) has infinitely many solutions say Φ (x,y,c)=0,c being an arbitrary constant such that through each point of S, one and only one member of the family Φ (x,y, c) = 0 passes.
Differential Equations Introduction Variables Separable
If the differential equation \(\frac{d y}{d x}=f(x, y)\) can be expressed in the form \(\frac{d y}{d x}=\frac{f(x)}{g(y)}\) or \(\frac{d y}{d x}=f(x) \cdot g(y)\) where f and g are continuous functions of a single variable, it is said to be of the form variables separable.
Differential Equations Introduction Working rule to find the General Solution (G. S.)
1. The given equation \(\frac{d y}{d x}=\frac{f(x)}{g(y)}\) can be written by separating variables as f(x) d x=g(y) d y.
2. Integrate both sides of (1) and add an arbitrary constant of integration to any one of the two sides.
3. General solution of (1) is \(\int f(x) d x=\int g(y) d y+c\) or \(\phi(x, y, c)=0\).
Note 1. The constant of integration c can be selected in any suitable form as \(c / 2, \log c, \sin c, e^c\) or \(\text{Tan}^{-1} c\) etc.
4. Some differential equations can be brought to variables’ separable form by some substitution
Consider \(\frac{d y}{d x}=f(a x+b y+c)\) ………………….(1)
Put \(a x+b y+c=u \Rightarrow \frac{d u}{d x}=a+b \frac{d y}{d x} \Rightarrow \frac{d y}{d x}=\frac{1}{b}\left(\frac{d u}{d x}-a\right)\) …………………….(2)
From (1) and (2), we have : \(\frac{1}{b}\left(\frac{d u}{d x}-a\right)=f(u) \Rightarrow d u=[a+b f(u)] d x\)
Separating the variables, we get: \(\frac{d u}{a+b f(u)}=d x\)
⇒ \(\int \frac{d u}{a+b f(u)}=\int d x+c_1\)
∴ The general solution of (1) is \(\int \frac{d u}{a+b f(u)}=\int d x+c_1\) where \(\mathbf{c}_1\) is an arbitrary constant or \(\phi\left(u, x, \mathrm{c}_1\right)=0 \text { or } \phi\left(a x+b y+c, x, \mathrm{c}_1\right)=0\)