Separable differential equation Calculator

Ask and get solution to your homeworkAsk now and get step-by-step solutions

Example

Created on 2024-06-20Asked by Penelope Harris (Solvelet student)

Solve the separable differential equation

\frac{dy}{dx} = xy

with the initial condition

y(0) = 1

Solution

To solve the separable differential equation

\frac{dy}{dx} = xy

with the initial condition

y(0) = 1

: 1. **Separate the variables:**

\frac{dy}{y} = x \, dx.

2. **Integrate both sides:**

\int \frac{1}{y} \, dy = \int x \, dx,

\ln|y| = \frac{x^2}{2} + C.

3. **Solve for

y

:**

y = e^{\frac{x^2}{2} + C} = e^C \cdot e^{\frac{x^2}{2}}.

Let

e^C = C_1

, then:

y = C_1 e^{\frac{x^2}{2}}.

4. **Apply the initial condition

y(0) = 1

:**

1 = C_1 e^0 \implies C_1 = 1.

5. **Solution:**

y = e^{\frac{x^2}{2}}.

Therefore, the solution to the differential equation is

y = e^{\frac{x^2}{2}}

. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Jackson Lewis on Solvelet

1. Solve the initial value problem

y' = 2x + 3y

y(0) = 1

.2. Determine the solution of the differential equation

(dy/dx)^2 = y

with initial condition

y(0) = 4

DefinitionA separable equation has the form g(y)dy=f(x)dx, So the variables are on opposite sides of the equation. The equation xdx = ydy is also an instance of a separable equation

Related Calculators

Division Calculator Simplify trigonometric expressions Calculator Binomial Distribution Calculator Taylor Series Calculator

Need topic explanation ? Get video explanation