Solve y=exp(3/x^3-2/x^2)

$$y = \exp ( \frac { 3 } { x ^ { 3 } } - \frac { 2 } { x ^ { 2 } } )$$

Solve for p (complex solution)

$$\left\{\begin{matrix}p=-\frac{yx^{2}}{e\left(2x-3\right)}\text{, }&x\neq \frac{3}{2}\text{ and }x\neq 0\\p\in \mathrm{C}\text{, }&x=\frac{3}{2}\text{ and }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x^{3}, the least common multiple of x^{3},x^{2}.

$$yx^{3}=exp\left(\frac{3}{x^{3}}-\frac{2}{x^{2}}\right)x^{3}$$

To multiply powers of the same base, add their exponents. Add 1 and 3 to get 4.

$$yx^{3}=ex^{4}p\left(\frac{3}{x^{3}}-\frac{2}{x^{2}}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x^{3} and x^{2} is x^{3}. Multiply \frac{2}{x^{2}} times \frac{x}{x}.

$$yx^{3}=ex^{4}p\left(\frac{3}{x^{3}}-\frac{2x}{x^{3}}\right)$$

Since \frac{3}{x^{3}} and \frac{2x}{x^{3}} have the same denominator, subtract them by subtracting their numerators.

$$yx^{3}=ex^{4}p\times \frac{3-2x}{x^{3}}$$

Express e\times \frac{3-2x}{x^{3}} as a single fraction.

$$yx^{3}=\frac{e\left(3-2x\right)}{x^{3}}x^{4}p$$

Use the distributive property to multiply e by 3-2x.

$$yx^{3}=\frac{3e-2ex}{x^{3}}x^{4}p$$

Express \frac{3e-2ex}{x^{3}}x^{4} as a single fraction.

$$yx^{3}=\frac{\left(3e-2ex\right)x^{4}}{x^{3}}p$$

Cancel out x^{3} in both numerator and denominator.

$$yx^{3}=x\left(-2ex+3e\right)p$$

Swap sides so that all variable terms are on the left hand side.

$$x\left(-2ex+3e\right)p=yx^{3}$$

Use the distributive property to multiply x by -2ex+3e.

$$\left(-2ex^{2}+3xe\right)p=yx^{3}$$

The equation is in standard form.

$$\left(3ex-2ex^{2}\right)p=yx^{3}$$

Divide both sides by -2ex^{2}+3xe.

$$\frac{\left(3ex-2ex^{2}\right)p}{3ex-2ex^{2}}=\frac{yx^{3}}{3ex-2ex^{2}}$$

Dividing by -2ex^{2}+3xe undoes the multiplication by -2ex^{2}+3xe.

$$p=\frac{yx^{3}}{3ex-2ex^{2}}$$

Divide yx^{3} by -2ex^{2}+3xe.

$$p=\frac{yx^{2}}{e\left(3-2x\right)}$$

Solve for p

$$\left\{\begin{matrix}p=-\frac{yx^{2}}{e\left(2x-3\right)}\text{, }&x\neq \frac{3}{2}\text{ and }x\neq 0\\p\in \mathrm{R}\text{, }&x=\frac{3}{2}\text{ and }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x^{3}, the least common multiple of x^{3},x^{2}.

$$yx^{3}=exp\left(\frac{3}{x^{3}}-\frac{2}{x^{2}}\right)x^{3}$$

To multiply powers of the same base, add their exponents. Add 1 and 3 to get 4.

$$yx^{3}=ex^{4}p\left(\frac{3}{x^{3}}-\frac{2}{x^{2}}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x^{3} and x^{2} is x^{3}. Multiply \frac{2}{x^{2}} times \frac{x}{x}.

$$yx^{3}=ex^{4}p\left(\frac{3}{x^{3}}-\frac{2x}{x^{3}}\right)$$

Since \frac{3}{x^{3}} and \frac{2x}{x^{3}} have the same denominator, subtract them by subtracting their numerators.

$$yx^{3}=ex^{4}p\times \frac{3-2x}{x^{3}}$$

Express e\times \frac{3-2x}{x^{3}} as a single fraction.

$$yx^{3}=\frac{e\left(3-2x\right)}{x^{3}}x^{4}p$$

Use the distributive property to multiply e by 3-2x.

$$yx^{3}=\frac{3e-2ex}{x^{3}}x^{4}p$$

Express \frac{3e-2ex}{x^{3}}x^{4} as a single fraction.

$$yx^{3}=\frac{\left(3e-2ex\right)x^{4}}{x^{3}}p$$

Cancel out x^{3} in both numerator and denominator.

$$yx^{3}=x\left(-2ex+3e\right)p$$

Swap sides so that all variable terms are on the left hand side.

$$x\left(-2ex+3e\right)p=yx^{3}$$

Use the distributive property to multiply x by -2ex+3e.

$$\left(-2ex^{2}+3xe\right)p=yx^{3}$$

The equation is in standard form.

$$\left(3ex-2ex^{2}\right)p=yx^{3}$$

Divide both sides by -2ex^{2}+3xe.

$$\frac{\left(3ex-2ex^{2}\right)p}{3ex-2ex^{2}}=\frac{yx^{3}}{3ex-2ex^{2}}$$

Dividing by -2ex^{2}+3xe undoes the multiplication by -2ex^{2}+3xe.

$$p=\frac{yx^{3}}{3ex-2ex^{2}}$$

Divide yx^{3} by -2ex^{2}+3xe.

$$p=\frac{yx^{2}}{e\left(3-2x\right)}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=-\frac{\sqrt{e}\left(\sqrt{p\left(3y+ep\right)}+\sqrt{e}p\right)}{y}\text{; }x=-\frac{\sqrt{e}\left(-\sqrt{p\left(3y+ep\right)}+\sqrt{e}p\right)}{y}\text{, }&y\neq 0\text{ and }p\neq 0\\x=\frac{3}{2}\text{, }&y=0\text{ and }p\neq 0\\x\neq 0\text{, }&y=0\text{ and }p=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=-\frac{\sqrt{e}\left(\sqrt{p\left(3y+ep\right)}+\sqrt{e}p\right)}{y}\text{; }x=-\frac{\sqrt{e}\left(-\sqrt{p\left(3y+ep\right)}+\sqrt{e}p\right)}{y}\text{, }&\left(p\neq 0\text{ and }y=-\frac{ep}{3}\right)\text{ or }\left(y\neq 0\text{ and }y\geq -\frac{ep}{3}\text{ and }p>0\right)\text{ or }\left(y\neq 0\text{ and }y\leq -\frac{ep}{3}\text{ and }p<0\right)\\x=\frac{3}{2}\text{, }&y=0\text{ and }p\neq 0\\x\neq 0\text{, }&y=0\text{ and }p=0\end{matrix}\right.$$