Answer to Question #292771 in Differential Equations for luna

Question

Solve the first order linear inhomogeneous differential equation using the constant variation method

xy^,-2y=2x⁴

Answer 1

Given a first order inhomogeneous linear differential equation of the form;

$\displaystyle y\prime+p(x)y=f(x)\cdots\cdots\cdots\color{red}(1)$

using constant variation method, the general solution is given by;

$\displaystyle y(x)=v(x)e^{q(x)}+ce^{q(x)}$

where $\displaystyle v\prime(x)=e^{-q(x)}f(x)$, $\displaystyle q(x)=\int[-p(x)]\ dx$, and $c$ is an arbitrary constant.

Now, the given DE is of the form;

$y\prime-\frac{2}{x}y=2x^3$

By comparing the given DE with $\color{red}(1)$;

$\displaystyle q(x)=\int[-p(x)]\ dx=\int\frac{2}{x}\ dx=2\ln x=\ln x^2$

$\displaystyle v\prime(x)=e^{-q(x)\ dx}\times f(x)=e^{-\ln x^2}\times2x^3=\frac{1}{x^2}\times2x^3=2x$

$\displaystyle \Rightarrow v(x)=\int2x\ dx=x^2$

Thus, the general solution is;

$\displaystyle y(x)=v(x)e^{q(x)}+ce^{q(x)}=x^2e^{\ln x^2}+ce^{\ln{x^2}}=x^4+cx^2$