“Separate first, always”
Which method to try
If the equation can be written with all the y terms on one side and all the x terms on the other, the rest is just two integrations.
Mathematics, Class 11 and 12 · Gujarat Board, CBSE, GUJCET and JEE
Differential Equations formulas
A short, highly patterned chapter. Identify the type, apply its method, and the marks follow. Variable separable and linear equations between them cover the great majority of what is asked.
13 formulas, 3 memory hooks, and the 5 mistakes that cost the most marks.
Basics
| What | Formula | Symbols and notes |
|---|---|---|
| Order | the highest derivative present | y'' gives order 2 |
| Degree | the power of the highest derivative, once free of radicals | not defined if it cannot be made polynomial |
| General solution | contains as many arbitrary constants as the order | a first order equation has one |
| Particular solution | the constants fixed by given conditions | substitute the initial values |
Methods
| What | Formula | Symbols and notes |
|---|---|---|
| Variable separable | write as f(y)dy = g(x)dx, then integrate both sides | the first thing to try |
| Homogeneous | substitute y = vx, so dy/dx = v + x dv/dx | when every term has the same total degree |
| Linear in y | dy/dx + Py = Q, with P and Q functions of x | the standard linear form |
| Integrating factor | IF = e∫P dx | multiply through by it |
| Linear solution | y × IF = ∫(Q × IF)dx + c | the result worth memorising |
| Linear in x | dx/dy + Px = Q | the same method with the roles swapped |
Forming equations
| What | Formula | Symbols and notes |
|---|---|---|
| From a family of curves | differentiate as many times as there are constants | then eliminate them |
| Growth and decay | dy/dt = ky gives y = y₀ekt | k positive for growth, negative for decay |
| Newton's law of cooling | dT/dt = −k(T − T₀) | the same shape of equation |
Taught by Darshit Bhatt, founder and lead faculty at Darshit Sir Institute. Instrumentation and Control engineer, with a master's degree and a chartered degree, teaching in Paldi, Ahmedabad since 2006.
Know the formula but not when to use it?
Darshit Sir has taught this in Paldi since 2006, in the classroom and live online.
Memory hooks
How to remember this chapter.
“e to the integral of P”
The integrating factor
Get the equation into dy/dx + Py = Q form first, with the coefficient of dy/dx equal to one, then IF = e∫P dx.
“One constant per order”
How many constants to expect
A first order equation has one arbitrary constant in its general solution, a second order two. Use it as a check.
Where marks go
The mistakes that cost the most.
- Finding the integrating factor before making the coefficient of dy/dx equal to one.
- Confusing order with degree. Order counts derivatives; degree is a power.
- Forgetting the constant of integration, which makes a general solution particular by accident.
- Substituting y = vx for an equation that is not homogeneous.
- Leaving the answer in terms of v after a homogeneous substitution instead of returning to y.
Test yourself
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