Solve the differential equation dy/dx=x/(t^3)+x

 Exercise

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

 

Multiply both sides of the equation by $dx$

$dy=\left(\frac{x}{t^3}+x\right)dx$

 

Integrate both sides of the differential equation, the left side with respect to $y$, and the right side with respect to $x$

$\int1dy=\int\left(\frac{x}{t^3}+x\right)dx$

 

Expand the integral $\int\left(\frac{x}{t^3}+x\right)dx$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately

$\int1dy=\int\frac{x}{t^3}dx+\int xdx$

 

Solve the integral $\int1dy$ and replace the result in the differential equation

$y=\int\frac{x}{t^3}dx+\int xdx$

 

Solve the integral $\int\frac{x}{t^3}dx+\int xdx$ and replace the result in the differential equation

$y=\frac{x^2}{2t^3}+\frac{1}{2}x^2+C_0$

$y=\frac{x^2}{2t^3}+\frac{1}{2}x^2+C_0$

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