x d x + y d y = 0
Solve for d (complex solution)
\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{C}\text{, }&x=iy\text{ or }x=-iy\end{matrix}\right.
Solve for d
\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&x=0\text{ and }y=0\end{matrix}\right.
Solve for x (complex solution)
\left\{\begin{matrix}\\x=iy\text{; }x=-iy\text{, }&\text{unconditionally}\\x\in \mathrm{C}\text{, }&d=0\end{matrix}\right.
Solve for x
\left\{\begin{matrix}x=0\text{, }&y=0\\x\in \mathrm{R}\text{, }&d=0\end{matrix}\right.
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x^{2}d+ydy=0
Multiply x and x to get x^{2}.
x^{2}d+y^{2}d=0
Multiply y and y to get y^{2}.
\left(x^{2}+y^{2}\right)d=0
Combine all terms containing d.
d=0
Divide 0 by x^{2}+y^{2}.
x^{2}d+ydy=0
Multiply x and x to get x^{2}.
x^{2}d+y^{2}d=0
Multiply y and y to get y^{2}.
\left(x^{2}+y^{2}\right)d=0
Combine all terms containing d.
d=0
Divide 0 by x^{2}+y^{2}.
Examples
Quadratic equation
{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
\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}