Question

Solve the following differential equation.

\( (x+2y^3)dy=ydy \)

Answer 1

Concept :

A differential equation is only said to be linear if dependent variables & its derivatives appears once in \( 1^{st} \) degree. and the form of equation is 

\( \frac{dx}{dy}+ Px=Q \)             .................(1)

Where, P and Q are functions of y or any integers.

Solution of linear differential equation is given by,

\( xIF=\int IF \times Qdy+C \),

Where,  \( I.F= e^{\int pdy} \)

 

So the given differential is linear differential equation this can be convert into above mentioned manner.

 

Step 1: Convert the given linear differential equation convert into as mentioned above. 

\( \frac{dx}{dy} = \frac{x+2y^2}{y} \\ \)

\( \frac{dx}{dy} -\frac{x}{y}=2y^2 \)             .................(2)

 

Step 2: Comparing the differential equation (2) with equation (1), we get

\( P=-\frac{x}{y} \)    ,      \( Q=2y^2 \)

 

Step 3: Find I.F value. 

I.F\( =e^{{\int -\frac{1}{y}} dy} =e^{-logy} \)

\( =e^{{(logy)}^{-1}}=\frac{1}{y} \)

 

Step 4: Substituting all values to find the Solution .

\( \frac{x}{y}=\int \frac{1}{y} 2y^2dy+C \)

\( \frac{x}{y}=2\int ydy+C \)

\( \frac{x}{y}=y^2+C \)

This is the required Solution.

 

This kind of differential question can be solve by linear differential equation method.