# What is a solution to the differential equation sqrtx+sqrtydy/dx with y(1)=4?

Jul 25, 2016

${x}^{\frac{3}{2}} + {y}^{\frac{3}{2}} = 9$.

#### Explanation:

Rewriting the Diff. Eqn. as, $\sqrt{x} \mathrm{dx} + \sqrt{y} \mathrm{dy} = 0$, we notice that it is a Separable Variable type Diff. Eqn.

To find its General Soln. , we integrate term-wise , i.e.,

$\int \sqrt{x} \mathrm{dx} + \int \sqrt{y} \mathrm{dy} = C$

$\therefore {x}^{\frac{3}{2}} / \left(\frac{3}{2}\right) + {y}^{\frac{3}{2}} / \left(\frac{3}{2}\right) = C$. or,

${x}^{\frac{3}{2}} + {y}^{\frac{3}{2}} = \frac{3}{2} \cdot C \ldots \ldots . \left(1\right)$

To determine $C$, we use the given cond. - called The Initial Cond. - that $y \left(1\right) = 4$, meaning, when $x = 1 , y = 4$.

Sub.ing in $\left(1\right)$, we get, $1 + {4}^{\frac{3}{2}} = \frac{3}{2} \cdot C \Rightarrow \frac{3}{2} \cdot C = 9$

Sub.ing $\frac{3}{2} \cdot C = 9$ in $\left(1\right)$, we get the complete Soln. - called The

Particular Soln. as ${x}^{\frac{3}{2}} + {y}^{\frac{3}{2}} = 9$.