Solve dy/dt=3at/1+t^2

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

Solve for a

$$\left\{\begin{matrix}\\a=0\text{, }&\text{unconditionally}\\a\in \mathrm{R}\text{, }&t=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by t^{2}+1.

$$\left(t^{2}+1\right)\frac{\mathrm{d}}{\mathrm{d}t}(y)=3at$$

Use the distributive property to multiply t^{2}+1 by \frac{\mathrm{d}}{\mathrm{d}t}(y).

$$t^{2}\frac{\mathrm{d}}{\mathrm{d}t}(y)+\frac{\mathrm{d}}{\mathrm{d}t}(y)=3at$$

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

$$3at=t^{2}\frac{\mathrm{d}}{\mathrm{d}t}(y)+\frac{\mathrm{d}}{\mathrm{d}t}(y)$$

The equation is in standard form.

$$3ta=0$$

Divide 0 by 3t.

$$a=0$$

Solve for t

$$\left\{\begin{matrix}\\t=0\text{, }&\text{unconditionally}\\t\in \mathrm{R}\text{, }&a=0\end{matrix}\right.$$