Question

Solve $ \frac{dy}{dx}+\frac{y}{x}={{x}^{3}} $

• A. $ y=\frac{{{x}^{4}}}{5}+Cx $
• B. $ y=\frac{{{x}^{3}}}{3}+Cx $
• C. $ y=\frac{{{x}^{3}}}{3}+C $
• D. $ y=\frac{{{x}^{4}}}{5}+\frac{C}{x} $

Answer 1

The Correct Option is D

We have, \(\frac{dy}{dx}+\frac{y}{x}={{x}^{3}}\) 
On comparing with \(\frac{dy}{dx}+py=Q,\) 
we get \(P=\frac{1}{x},Q={{x}^{3}}\)
\(\therefore\) \(IF={{e}^{\int{\frac{1}{x}dx}}}={{e}^{{{\log }_{e}}x}}=x\) 
Now, solution of given differential equation is 
\(y.If=\int{!\,IF\,dx+C}\)
\(\Rightarrow\) \(yx=\int{{{x}^{3}}.x\,dx+C}\)
\(\Rightarrow\) \(yx=\int{{{x}^{4}}\,dx+C}\)
\(\Rightarrow\) \(yx=\frac{{{x}^{5}}}{5}+C\)
\(\Rightarrow\) \(y=\frac{{{x}^{4}}}{5}+\frac{C}{x}\)

In mathematics, a differential equation is an equation that defines a relationship between functions and their derivatives. A differential equation establishes a connection between the rate of change of physical values, which are represented by derivatives, and their function-based representation. Finding the collection of values that fulfil the equation and researching them and their attributes make up differential equations. Using formulae, it is possible to solve simple differential equations. When there is no existing solution to a difficult equation, computers are frequently utilised to find one.

A differential equation is one that combines terms (one or more) and the derivatives of one dependent variable with the other independent variable.

dy/dx = f(x)

Order of a Differential Equation is defined as the ‘order of the highest order’ derivatives of the dependent variable with respect to the independent variable in the differential equation.

dy/dx = e^x

(highest derivative of first order)

d²y/dx² + y = 0

(highest derivative of second order)

d³y/dx³ + x² (d²y/dx²)³ = 0

(highest derivative of third order)