Differential equations are classified into different types such as Linear differential equations, exact and non-exact differential equations, homogeneous differential equations etc. All such equations can be solved by using various methods.

Certain non-exact differential equations can be directly solved by careful inspection of them. The method used for solving such equations is known as the ‘Method of inspection’.

**What is an exact differential equation? –**

The equation of type A dp+B dq are called exact differential equations only if they satisfy the condition A/q=B/p.

In this equation p and q are variables and A(p,q) & B(p,q) are two functions.

**What is a non-exact differential equation? – **

The equation of type A dp+B dq are called non-exact differential equations only if they satisfy the condition A/qB/p.

In this equation p and q are variables and A(p,q) & B(p,q) are two functions.

**How to solve a non-exact differential equation?-**

Generally, non-exact differential equations are first converted into exact differential equations, and then they are solved. But sometimes, it becomes very difficult to convert a given non-exact differential equation into an exact differential equation. Hence, we can use the ‘Method of inspection’ to solve such non-exact equations in such cases.

Before understanding the ‘Method of inspection,’ we first need to learn the concept of ‘Integrable’.

**Integrables –**

Integrables are parts of the differential equations whose integration can be done easily.

E.g. d(p2)=p2 , d(pq)=pq etc.

Here is the sign of integration (antiderivative), and d() represents the differential of a function in the argument.

**Some important integrables –**

**q dp+p dq=d(pq)****(q dp-p dq)/****q****2****=d(p/q)****dp+dq=d(p+q)****d(q/p)=(p dq-q dp)/****p****2****d(****p****2****/q)=(2pqdp-****p****2****dq)/****q****2****d(****q****2****/p)=(2pqdq-****q****2****dp)/****p****2****d(****p****2****/****q****2****)=(2p****q****2****dp-2****p****2****qdq)/****q****4****d(****q****2****/****p****2****)=(2q****p****2****dq-2****q****2****pdp)/****p****4****d(log(pq))=(p dq+q dp)/pq****d(log(p/q))=(q dp-p dq)/pq****d(log(q/p))=(p dq-q dp)/pq****d(log(p+q))=(dp+dq)/p+q****d(log(****p****2****+****q****2****))=(p dp+q dq)/(****p****2****+****q****2****)****d(ta****n****-1****(q/p))=(p dq-q dp)/(****p****2****+****q****2****)****d(ta****n****-1****(p/q))=(q dp-p dq)/(****p****2****+****q****2****)****d(-1/pq)=(p dq-q dp)/****p****2****q****2****d(****e****p****/q)=(q****e****p****dq-****e****p****dq)/****q****2****d(****e****q****/p)=(p****e****q****dp-****e****q****dp)/****p****2****d(****p****2****+****q****2****)=(p dp+q dq)/****p****2****+****q****2****d(****p****m****,****q****n****)=****p****m-1****q****n-1****(mq dp+np dq)****d[(1/2)*(log (p+q)/(p-q))]=(p dq-q dp)/(****p****2****–****q****2****)****d[f(p,q****)****1-n****]/(1-n)=f'(p,q)/[f(p,q****)****n****]****d(1/q-1/p)=d(1/q)-d(1/p)=dp/****p****2****-dq/****q****2**

Remembering these integrables helps us a lot while solving problems.

**Procedure to solve the problem –**

**Step 1 – **

Check whether the given differential equation is exact or not. If it is non-exact, then only proceed with the next step.

**Step 2 –**

Rearrange the given differential equation so that the integrables are visible.

**Step 3 – **

Integrate the entire equation to get the final answer.

Let us understand this method with an example.

**Solve:** p dq-q dp=(p2+q2)dp

**Solution: **We first check whether the given equation is exact or not.

Given differential equation is

p dq-q dp=(p2+q2)dp …………..(1)

∴ (-q-q2–p2)dp+p dq=0

Comparing this equation with the standard form A dp+B dq=0, we get

A=(-q-q2–p2) & B=p

A/q=(-q-q2–p2)/q=-1-2q

B/p=1

∴ A/qB/p

Therefore, the given differential equation is not exact.

Now, from equation (1) we can write

p dq-q dp=(p2+q2)dp

Dividing both sides by (p2+q2), we get

(p dq-q dp)/(p2+q2)=(p2+q2)dp/(p2+q2)

(p dq-q dp)/(p2+q2)=dp ………..(2)

But as we have seen that

(p dq-q dp)/(p2+q2)=d(tan-1(q/p))

Therefore, equation (2) becomes

∴ d (tan-1(q/p))-dp=0…………….(3)

Equation (3) contains an easy-to-solve integrable. By integrating equation (2), we can get the solution for the given differential equation. Thus, integrating both sides of equation (3) we get

d (tan-1(q/p))-dp=0

∴ d(tan-1(q/p))-dp=0

∴ tan-1(q/p)+A1-p+A2=0

∴ tan-1(q/p)=p+A

Where, A1, A2,A are arbitrary constants.

This is the required solution.

**Advantages of inspection method –**

- It is an easy method.
- It does not include conversion of non-exact to exact differential equations.

**Disadvantages of inspection method – **

- One might find it difficult to remember all the integrables.

**Conclusion – **

In this article, we studied the method of inspection to solve non-exact differential equations. We have also solved one problem with the help of the inspection method.