# Solution of Differential Equation

Here you will learn how to find solution of differential equation i.e. general solution and particular solution with examples.

Let’s begin –

## Solution of Differential Equation

The solution of differential equation is a relation between the variables involved which satisfies the differential equation.

for example, y = $$e^x$$ is a solution of the differential equations $$dy\over dx$$ = y.

#### General Solution

The solution which contains as many as arbitrary constants as the order of the differential equations is called the general solution.

for example, y = A cos x + B sin x is the general solution of the equation $$d^2y\over dx^2$$ + y = 0.

#### Particular Solution

The solution obtained by giving particular values to the arbitrary constants in the general solution of a differential equations is called a particular solution.

for example, y = 3 cos x + 2 sin x is the particular solution of the equation $$d^2y\over dx^2$$ + y = 0.

Example : Show that y = Ax  + $$B\over x$$, x $$\ne$$ 0 is a solution of the differential equation $$x^2$$$$d^2y\over dx^2$$ + x$$dy\over dx$$ – y = 0

Solution : We have,

y = Ax  + $$B\over x$$ = 0, x $$\ne$$ 0                ……..(i)

Differentiating both sides with respect to x, we get

$$dy\over dx$$ = A – $$B\over x^2$$                   ……….(ii)

Again differentiating with respect to x, we get

$$d^2y\over dx^2$$ = $$2B\over x^3$$            …………..(iii)

Substituting the values of y, $$dy\over dx$$ and $$d^2y\over dx^2$$ in $$x^2$$$$d^2y\over dx^2$$ + x$$dy\over dx$$ – y , we get

$$x^2$$$$d^2y\over dx^2$$ + x$$dy\over dx$$ – y = $$x^2$$($$2B\over x^3$$) + x(A – $$B\over x^2$$) – (Ax + $$B\over x$$)

= $$2B\over x$$ + Ax – $$B\over x$$ – Ax – $$B\over x$$ = 0

Thus, the function y = Ax  + $$B\over x$$ satisfies the given equation.