( 1 - \frac { y } { x ^ { 2 } } e ^ { y / x } ) d x + ( 1 + \frac { 1 } { x } e ^ { y / x } ) d y = 0
Solve for d (complex solution)
\left\{\begin{matrix}d=0\text{, }&x\neq 0\\d\in \mathrm{C}\text{, }&x=-y\text{ and }y\neq 0\end{matrix}\right.
Solve for d
\left\{\begin{matrix}d=0\text{, }&x\neq 0\\d\in \mathrm{R}\text{, }&x=-y\text{ and }y\neq 0\end{matrix}\right.
Solve for x
\left\{\begin{matrix}x=-y\text{, }&y\neq 0\\x\neq 0\text{, }&d=0\end{matrix}\right.
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\left(1-\frac{y}{x^{2}}e^{\frac{y}{x}}\right)dxx^{2}+\left(1+\frac{1}{x}e^{\frac{y}{x}}\right)dyx^{2}=0
Multiply both sides of the equation by x^{2}, the least common multiple of x^{2},x.
\left(1-\frac{y}{x^{2}}e^{\frac{y}{x}}\right)dx^{3}+\left(1+\frac{1}{x}e^{\frac{y}{x}}\right)dyx^{2}=0
To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.
\left(1-\frac{y}{x^{2}}e^{\frac{y}{x}}\right)dx^{3}+\left(d+\frac{1}{x}e^{\frac{y}{x}}d\right)yx^{2}=0
Use the distributive property to multiply 1+\frac{1}{x}e^{\frac{y}{x}} by d.
\left(1-\frac{y}{x^{2}}e^{\frac{y}{x}}\right)dx^{3}+\left(d+\frac{d}{x}e^{\frac{y}{x}}\right)yx^{2}=0
Express \frac{1}{x}d as a single fraction.
\left(1-\frac{y}{x^{2}}e^{\frac{y}{x}}\right)dx^{3}+\left(dy+\frac{d}{x}e^{\frac{y}{x}}y\right)x^{2}=0
Use the distributive property to multiply d+\frac{d}{x}e^{\frac{y}{x}} by y.
\left(1-\frac{y}{x^{2}}e^{\frac{y}{x}}\right)dx^{3}+\left(dy+\frac{dy}{x}e^{\frac{y}{x}}\right)x^{2}=0
Express \frac{d}{x}y as a single fraction.
\left(1-\frac{y}{x^{2}}e^{\frac{y}{x}}\right)dx^{3}+dyx^{2}+\frac{dy}{x}e^{\frac{y}{x}}x^{2}=0
Use the distributive property to multiply dy+\frac{dy}{x}e^{\frac{y}{x}} by x^{2}.
\left(1-\frac{y}{x^{2}}e^{\frac{y}{x}}\right)dx^{3}+dyx^{2}+\frac{dyx^{2}}{x}e^{\frac{y}{x}}=0
Express \frac{dy}{x}x^{2} as a single fraction.
\left(1-\frac{y}{x^{2}}e^{\frac{y}{x}}\right)dx^{3}+dyx^{2}+dxye^{\frac{y}{x}}=0
Cancel out x in both numerator and denominator.
\left(1-\frac{y}{x^{2}}e^{\frac{y}{x}}\right)dx^{3}x^{2}+dyx^{2}x^{2}+dxye^{\frac{y}{x}}x^{2}=0
Multiply both sides of the equation by x^{2}.
\left(1-\frac{y}{x^{2}}e^{\frac{y}{x}}\right)dx^{3}x^{2}x^{2}+dyx^{2}x^{2}x^{2}+dxye^{\frac{y}{x}}x^{2}x^{2}=0
Multiply both sides of the equation by x^{2}.
\left(1-\frac{y}{x^{2}}e^{\frac{y}{x}}\right)dx^{5}x^{2}+dyx^{2}x^{2}x^{2}+dxye^{\frac{y}{x}}x^{2}x^{2}=0
To multiply powers of the same base, add their exponents. Add 3 and 2 to get 5.
\left(1-\frac{y}{x^{2}}e^{\frac{y}{x}}\right)dx^{7}+dyx^{2}x^{2}x^{2}+dxye^{\frac{y}{x}}x^{2}x^{2}=0
To multiply powers of the same base, add their exponents. Add 5 and 2 to get 7.
\left(1-\frac{y}{x^{2}}e^{\frac{y}{x}}\right)dx^{7}+dyx^{4}x^{2}+dxye^{\frac{y}{x}}x^{2}x^{2}=0
To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.
\left(1-\frac{y}{x^{2}}e^{\frac{y}{x}}\right)dx^{7}+dyx^{6}+dxye^{\frac{y}{x}}x^{2}x^{2}=0
To multiply powers of the same base, add their exponents. Add 4 and 2 to get 6.
\left(1-\frac{y}{x^{2}}e^{\frac{y}{x}}\right)dx^{7}+dyx^{6}+dx^{3}ye^{\frac{y}{x}}x^{2}=0
To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.
\left(1-\frac{y}{x^{2}}e^{\frac{y}{x}}\right)dx^{7}+dyx^{6}+dx^{5}ye^{\frac{y}{x}}=0
To multiply powers of the same base, add their exponents. Add 3 and 2 to get 5.
\left(d-\frac{y}{x^{2}}e^{\frac{y}{x}}d\right)x^{7}+dyx^{6}+dx^{5}ye^{\frac{y}{x}}=0
Use the distributive property to multiply 1-\frac{y}{x^{2}}e^{\frac{y}{x}} by d.
\left(d-\frac{yd}{x^{2}}e^{\frac{y}{x}}\right)x^{7}+dyx^{6}+dx^{5}ye^{\frac{y}{x}}=0
Express \frac{y}{x^{2}}d as a single fraction.
dx^{7}-\frac{yd}{x^{2}}e^{\frac{y}{x}}x^{7}+dyx^{6}+dx^{5}ye^{\frac{y}{x}}=0
Use the distributive property to multiply d-\frac{yd}{x^{2}}e^{\frac{y}{x}} by x^{7}.
dx^{7}-\frac{ydx^{7}}{x^{2}}e^{\frac{y}{x}}+dyx^{6}+dx^{5}ye^{\frac{y}{x}}=0
Express \frac{yd}{x^{2}}x^{7} as a single fraction.
dx^{7}-dyx^{5}e^{\frac{y}{x}}+dyx^{6}+dx^{5}ye^{\frac{y}{x}}=0
Cancel out x^{2} in both numerator and denominator.
dx^{7}+dyx^{6}=0
Combine -dyx^{5}e^{\frac{y}{x}} and dx^{5}ye^{\frac{y}{x}} to get 0.
\left(x^{7}+yx^{6}\right)d=0
Combine all terms containing d.
d=0
Divide 0 by x^{7}+yx^{6}.
\left(1-\frac{y}{x^{2}}e^{\frac{y}{x}}\right)dxx^{2}+\left(1+\frac{1}{x}e^{\frac{y}{x}}\right)dyx^{2}=0
Multiply both sides of the equation by x^{2}, the least common multiple of x^{2},x.
\left(1-\frac{y}{x^{2}}e^{\frac{y}{x}}\right)dx^{3}+\left(1+\frac{1}{x}e^{\frac{y}{x}}\right)dyx^{2}=0
To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.
\left(1-\frac{y}{x^{2}}e^{\frac{y}{x}}\right)dx^{3}+\left(d+\frac{1}{x}e^{\frac{y}{x}}d\right)yx^{2}=0
Use the distributive property to multiply 1+\frac{1}{x}e^{\frac{y}{x}} by d.
\left(1-\frac{y}{x^{2}}e^{\frac{y}{x}}\right)dx^{3}+\left(d+\frac{d}{x}e^{\frac{y}{x}}\right)yx^{2}=0
Express \frac{1}{x}d as a single fraction.
\left(1-\frac{y}{x^{2}}e^{\frac{y}{x}}\right)dx^{3}+\left(dy+\frac{d}{x}e^{\frac{y}{x}}y\right)x^{2}=0
Use the distributive property to multiply d+\frac{d}{x}e^{\frac{y}{x}} by y.
\left(1-\frac{y}{x^{2}}e^{\frac{y}{x}}\right)dx^{3}+\left(dy+\frac{dy}{x}e^{\frac{y}{x}}\right)x^{2}=0
Express \frac{d}{x}y as a single fraction.
\left(1-\frac{y}{x^{2}}e^{\frac{y}{x}}\right)dx^{3}+dyx^{2}+\frac{dy}{x}e^{\frac{y}{x}}x^{2}=0
Use the distributive property to multiply dy+\frac{dy}{x}e^{\frac{y}{x}} by x^{2}.
\left(1-\frac{y}{x^{2}}e^{\frac{y}{x}}\right)dx^{3}+dyx^{2}+\frac{dyx^{2}}{x}e^{\frac{y}{x}}=0
Express \frac{dy}{x}x^{2} as a single fraction.
\left(1-\frac{y}{x^{2}}e^{\frac{y}{x}}\right)dx^{3}+dyx^{2}+dxye^{\frac{y}{x}}=0
Cancel out x in both numerator and denominator.
\left(1-\frac{y}{x^{2}}e^{\frac{y}{x}}\right)dx^{3}x^{2}+dyx^{2}x^{2}+dxye^{\frac{y}{x}}x^{2}=0
Multiply both sides of the equation by x^{2}.
\left(1-\frac{y}{x^{2}}e^{\frac{y}{x}}\right)dx^{3}x^{2}x^{2}+dyx^{2}x^{2}x^{2}+dxye^{\frac{y}{x}}x^{2}x^{2}=0
Multiply both sides of the equation by x^{2}.
\left(1-\frac{y}{x^{2}}e^{\frac{y}{x}}\right)dx^{5}x^{2}+dyx^{2}x^{2}x^{2}+dxye^{\frac{y}{x}}x^{2}x^{2}=0
To multiply powers of the same base, add their exponents. Add 3 and 2 to get 5.
\left(1-\frac{y}{x^{2}}e^{\frac{y}{x}}\right)dx^{7}+dyx^{2}x^{2}x^{2}+dxye^{\frac{y}{x}}x^{2}x^{2}=0
To multiply powers of the same base, add their exponents. Add 5 and 2 to get 7.
\left(1-\frac{y}{x^{2}}e^{\frac{y}{x}}\right)dx^{7}+dyx^{4}x^{2}+dxye^{\frac{y}{x}}x^{2}x^{2}=0
To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.
\left(1-\frac{y}{x^{2}}e^{\frac{y}{x}}\right)dx^{7}+dyx^{6}+dxye^{\frac{y}{x}}x^{2}x^{2}=0
To multiply powers of the same base, add their exponents. Add 4 and 2 to get 6.
\left(1-\frac{y}{x^{2}}e^{\frac{y}{x}}\right)dx^{7}+dyx^{6}+dx^{3}ye^{\frac{y}{x}}x^{2}=0
To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.
\left(1-\frac{y}{x^{2}}e^{\frac{y}{x}}\right)dx^{7}+dyx^{6}+dx^{5}ye^{\frac{y}{x}}=0
To multiply powers of the same base, add their exponents. Add 3 and 2 to get 5.
\left(d-\frac{y}{x^{2}}e^{\frac{y}{x}}d\right)x^{7}+dyx^{6}+dx^{5}ye^{\frac{y}{x}}=0
Use the distributive property to multiply 1-\frac{y}{x^{2}}e^{\frac{y}{x}} by d.
\left(d-\frac{yd}{x^{2}}e^{\frac{y}{x}}\right)x^{7}+dyx^{6}+dx^{5}ye^{\frac{y}{x}}=0
Express \frac{y}{x^{2}}d as a single fraction.
dx^{7}-\frac{yd}{x^{2}}e^{\frac{y}{x}}x^{7}+dyx^{6}+dx^{5}ye^{\frac{y}{x}}=0
Use the distributive property to multiply d-\frac{yd}{x^{2}}e^{\frac{y}{x}} by x^{7}.
dx^{7}-\frac{ydx^{7}}{x^{2}}e^{\frac{y}{x}}+dyx^{6}+dx^{5}ye^{\frac{y}{x}}=0
Express \frac{yd}{x^{2}}x^{7} as a single fraction.
dx^{7}-dyx^{5}e^{\frac{y}{x}}+dyx^{6}+dx^{5}ye^{\frac{y}{x}}=0
Cancel out x^{2} in both numerator and denominator.
dx^{7}+dyx^{6}=0
Combine -dyx^{5}e^{\frac{y}{x}} and dx^{5}ye^{\frac{y}{x}} to get 0.
\left(x^{7}+yx^{6}\right)d=0
Combine all terms containing d.
d=0
Divide 0 by x^{7}+yx^{6}.
Examples
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{ x } ^ { 2 } - 4 x - 5 = 0
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y = 3x + 4
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\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}