Differential Equation Formulas

The concept of differential equations is used in various fields of the real-world like physics, engineering, and economics. To make your calculations on Differential Equations easily use the provided list of Differential Equation formulas. Also, stick the Differential Equation Formula Sheet on the wall at study place and memorize them regularly.

1. Differential Equation

An equation containing an independent variable, dependent variable and differential coefficients of dependent variable with respect to independent variable is called a differential equation.

2. Order of Differential Equation

The order of a differential equation is the order of the highest derivative occurring in the differential equation.

3. Degree of Differential Equation

The degree of a differential equation is the degree of the highest order derivative when differential coefficients are free from radical and fractional power.

4. General solution

The solution which contains a number of arbitrary constants equal to the order of the equation is called general solution or complete integral or complete primitive of differential equation.

5. Differential Equations of the Form \(\frac{d y}{d x}\) = f(x)

∫ dy = ∫ f(x) dx + c

6. Differential Equations of the Form \(\frac{d y}{d x}\) = f(x) g(y)

(variable separable pattern)
∫ \(\frac{d y}{g(y)}\) = ∫ f(x) dx + c

7. Differential Equations of the Form of \(\frac{d y}{d x}\) = f (ax + by + c)

To solve this type of differential equations, we put ax + by + c = v and
\(\frac{d y}{d x}=\frac{1}{b}\left(\frac{d v}{d x}-a\right)\)
∴ \(\frac{d v}{a+b f(v)}\) = dx
So solution is by integrating ∫\(\frac{d v}{a+b f(v)}\) = ∫ dx

8. Differential Equation of Homogeneous Type or f(y/x) = 0 pattern

To solve the homogeneous differential equation \(\frac{d y}{d x}=\frac{f(x, y)}{g(x, y)}\),
substitute y = vx and so \(\frac{d y}{d x}\) = v + x \(\frac{d v}{d x}\)
Thus v + x \(\frac{d y}{d x}\) = f( v) ⇒ \(\frac{d x}{x}\) = \(\frac{d v}{f(v)-v}\)
Therefore solution is \(\int \frac{d x}{x}=\int \frac{d v}{f(v)-v}+c\)

9. Differential Equations reducible to Homogeneous Form

A differential equation of the form \(\frac{d y}{d x}=\frac{a_{1} x+b_{1} y+c_{1}}{a_{2} x+b_{2} y+c_{2}}\), where \(\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}\) can be reduced to homogeneous form by adopting the following procedure –
put x = X + h, y = Y + kso that = \(\frac{d Y}{d X}=\frac{d y}{d x}\)
The equation then transformed to
\(\frac{d Y}{d X}=\frac{a_{1} X+b_{1} Y+\left(a_{1} h+b_{1} k+c_{1}\right)}{a_{2} X+b_{2} Y+\left(a_{2} h+b_{2} k+c_{2}\right)}\)
Now choose h and k such that a₁h + b₁k + c₁ = 0 and a₂h + b₂k + c₂ = 0.
Then for these values of h and k the equation becomes \(\frac{d Y}{d X}=\frac{a_{1} X+b_{1} Y}{a_{2} X+b_{2} Y}\)
This is a homogeneous equation which can be solved by putting Y = vX and then Y and X should be replaced by y – k and x – h.

10. Linear Differential Equations

A differential equation is linear if the dependent variable (y) and its derivative appear only in first degree. The general form of a linear differential equation of first order is \(\frac{d y}{d x}\) + Py = Q
y e^∫Pdx = ∫Qe^∫Pdxdx + c
which is the required solution, where c is the constant and e^∫Pdx is called the integrating factor.

11. Equation reducible to linear form [Bernoulli’s equation]

If the given equation is of the form \(\frac{d y}{d x}p\) + P. f(y) = Q. g(y), where P
and Q are functions of x along, we divide the equation by g(y), we get
\(\frac{1}{g(y)} \frac{d y}{d x}+P \cdot \frac{f(y)}{g(y)}\)
Now substituted = \(\frac{f(y)}{g(y)}\) = v and solve same as method given in linear
differential equation.

12. Differential Equation in the form of \(\frac{d^{2} y}{d x^{2}}\) = f(x)
then its solution can be obtained by integrating it with respect to x twice.

13. Some important results (solution by inspection)

• d(xy) = xdy + ydx
• d\(\left(\frac{x}{y}\right)\) = \(\frac{y d x-x d y}{y^{2}}\), y ≠ 0
• d\(\left(\frac{y}{x}\right)\) = \(\frac{x d y-y d x}{x^{2}}\), x ≠ 0
• d\(\left(\frac{x^{2}}{y}\right)\) = \(\frac{2 x y d x-x^{2} d y}{y^{2}}\), y ≠ 0
• d\(\left(\frac{y^{2}}{x}\right)\) = \(\frac{2 x y d y-y^{2} d x}{x^{2}}\)
• d\(\left(\frac{x^{2}}{y^{2}}\right)\) = \(\frac{2 x y^{2} d x-2 x^{2} y d y}{y^{4}}\)
• d\(\left(\tan ^{-1}\left(\frac{y}{x}\right)\right)\) = \(\frac{x d y-y d x}{x^{2}+y^{2}}\)
• d(log_e(xy)) = \(\frac{x d y+y d x}{x y}\)
• d\(\left(\log _{e}\left(\frac{x}{y}\right)\right)\) = \(\frac{y d x-x d y}{x y}\)

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