Solve the differential equation dy/dx=3x-z

 Exercise

\frac{dy}{dx}=3x-z

 

Multiply both sides of the equation by $dx$

dy=\left(3x-z\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(3x-z\right)dx

 

Expand the integral $\int\left(3x-z\right)dx$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately

\int1dy=\int3xdx+\int-zdx

 

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

y=\int3xdx+\int-zdx

 

Solve the integral $\int3xdx+\int-zdx$ and replace the result in the differential equation

y=\frac{3}{2}x^2-zx+C_0

y=\frac{3}{2}x^2-zx+C_0

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