Formula sheet: Differential Equations

Differential Equations14 min readLesson 1 of 7

1.1 Order, degree and solutions Derivatives appear in these equations, and two labels describe them. The order is read from the highest derivative in the equation. The degree is the power of that highest derivative once the equation has been cleared of fractions and radicals and is polynomial in the derivatives. A solution with no arbitrary constant is a particular solution, and a solution carrying as many arbitrary constants as the order is the general solution. Verifying a solution means differentiating it enough times and substituting into the equation, rather than merely checking that it looks plausible. 1.2 The three solution methods Almost every question in this chapter is solved by one of three methods. If the variables can be separated, integrate each side separately. If the equation is homogeneous, substitute y equal to v x to reduce it to a separable equation in v and x. If the equation is linear in y, find the integrating factor and use the product rule backwards. Deciding which method applies is the whole exam skill; the integration after that belongs to the previous chapter.

order = order of the highest derivative, degree = power of the highest derivative

Definitions used in Exercise 9.1

\frac{d y}{d x} = f (x) g (y) ⟹ \int \frac{d y}{g ( y )} = \int f (x) d x

Variable separable method

homogeneous: substitute y = v x, \frac{d y}{d x} = v + x \frac{d v}{d x}

The substitution that turns a homogeneous equation into a separable one

\frac{d y}{d x} + P y = Q, IF = e^{\int P d x}

First order linear equation and its integrating factor

y \cdot IF = \int Q \cdot IF d x + C

Solution of the linear equation

\frac{d x}{d y} + P_{1} x = Q_{1}

Linear equation written with x as the unknown, used when y appears simply

1.3 Forming an equation A family of curves with one arbitrary constant gives a first order equation: differentiate the family once and eliminate the constant between the equation and its derivative. Two constants need two differentiations and the elimination of both constants. The number of constants equals the order of the resulting equation, which is a useful check on the answer. The integrating factor is never zero and is chosen so that the left side becomes the derivative of y times the factor. When the coefficient P is a function whose integral is a logarithm, the factor usually becomes a power or a trigonometric function, so simplify it before using it in the final formula.

Exam

State the method before starting: separable, homogeneous or linear. Then write the equation in the standard form of that method. A correct standard form with a small integration slip usually still earns most of the method marks.

Find the order and degree only after clearing fractions and radicals.
For a homogeneous equation, confirm that every term has the same total degree.
For a linear equation, write it as dy by dx plus Py equals Q before finding the integrating factor.
Use the initial condition only after the general solution is written.
Verify the final solution by differentiation when time allows.

1.4 Reading the sheet in the exam The sheet is arranged in the order in which the questions come. The first two formulas answer the order and degree part. The next three answer the method parts. The last two cover formation and the exchange of the roles of x and y. In the exam, copy the standard form of the chosen method onto the answer sheet before substituting anything; the rest of the solution is bookkeeping.

in hinglish

Differential equation mein pehla kaam order aur degree nikaalna hai: order sabse badi derivative ka order hai, degree us derivative ki power hai jab equation polynomial form mein ho. General solution mein order ke barabar arbitrary constants hote hain, aur initial condition se particular solution milta hai. Teen methods yaad rakho: variable separable mein variables alag karke integrate karo; homogeneous mein y = vx substitute karo; linear mein IF = e^(\intPdx) nikaal kar y\cdotIF = \intQ\cdotIF dx likho. Method pehle pehchano, phir integration karo, aur aakhir mein answer differentiate karke verify karo. Aur standard form likhna aadha kaam hai: pehle method decide karo, phir equation ko usi form mein likho. Initial condition ke baad hi constant nikaalo aur answer differentiate karke check karo.