Solve dt1/t^3+t^2+t+1=dx/x

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

Solve for d

$$\left\{\begin{matrix}d=0\text{, }&t\neq -1\text{ and }x\neq 0\\d\in \mathrm{R}\text{, }&x\neq 0\text{ and }t=-\sqrt[3]{\frac{\sqrt{93}}{18}+\frac{29}{54}}-\sqrt[3]{-\frac{\sqrt{93}}{18}+\frac{29}{54}}-\frac{1}{3}\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x\left(t+1\right)\left(t^{2}+1\right), the least common multiple of t^{3}+t^{2}+t+1,x.

$$dtx\times 1=\left(t+1\right)\left(t^{2}+1\right)dx$$

Use the distributive property to multiply t+1 by t^{2}+1.

$$dtx\times 1=\left(t^{3}+t+t^{2}+1\right)dx$$

Use the distributive property to multiply t^{3}+t+t^{2}+1 by d.

$$dtx\times 1=\left(t^{3}d+td+t^{2}d+d\right)x$$

Use the distributive property to multiply t^{3}d+td+t^{2}d+d by x.

$$dtx\times 1=t^{3}dx+tdx+t^{2}dx+dx$$

Subtract t^{3}dx from both sides.

$$dtx\times 1-t^{3}dx=tdx+t^{2}dx+dx$$

Subtract tdx from both sides.

$$dtx\times 1-t^{3}dx-tdx=t^{2}dx+dx$$

Subtract t^{2}dx from both sides.

$$dtx\times 1-t^{3}dx-tdx-t^{2}dx=dx$$

Subtract dx from both sides.

$$dtx\times 1-t^{3}dx-tdx-t^{2}dx-dx=0$$

Reorder the terms.

$$dtx-dxt^{3}-dtx-dxt^{2}-dx=0$$

Combine dtx and -dtx to get 0.

$$-dxt^{3}-dxt^{2}-dx=0$$

Combine all terms containing d.

$$\left(-xt^{3}-xt^{2}-x\right)d=0$$

Divide 0 by -xt^{3}-xt^{2}-x.

$$d=0$$

Solve for t

$$\left\{\begin{matrix}t=-\frac{2^{\frac{2}{3}}\sqrt[3]{3\sqrt{93}+29}}{6}-\frac{2^{\frac{2}{3}}\sqrt[3]{29-3\sqrt{93}}}{6}-\frac{1}{3}\text{, }&x\neq 0\\t\neq -1\text{, }&d=0\text{ and }x\neq 0\end{matrix}\right.$$