Mathematics - Differential Equation Question with Solution | TestHub

We have,$\frac{dy}{dx}+\frac{3y}{4x}=7$$($where$y=f\left(x\right))$

This is a linear differential equation.

Integrating factor,$I.F.$$=e^{\int \frac{3dx}{4x}}=e^{\frac{3}{4}lnx}=x^{\frac{3}{4}}$

So, integrating the given differential equation using$I.F.$

$\Rightarrow y.x^{\frac{3}{4}}=\int 7.x^{\frac{3}{4}}dx$

$\Rightarrow y.x^{\frac{3}{4}}=7.\frac{4x^{\frac{7}{4}}}{7}+C$

$\Rightarrow y\left(x^{\frac{3}{4}}\right)=4x^{\frac{7}{4}}+C$

$\Rightarrow f\left(x\right)=4x+C{x}^{\frac{-3}{4}}$

$\Rightarrow \underset{x\to 0^{+}}{\mathrm{l}\mathrm{i}\mathrm{m}}xf\left(\frac{1}{x}\right)\Rightarrow \underset{x\to 0^{+}}{\mathrm{l}\mathrm{i}\mathrm{m}}x\left(\frac{4}{x}+C{x}^{\frac{3}{4}}\right)=4$