(3y2−t2y5)dydt+t2y4=0 Y(1)=1

5 min read Jun 16, 2024

Solving the Differential Equation (3y^2 - t^2y^5)dy/dt + t^2y^4 = 0 with Initial Condition y(1) = 1

This problem involves solving a first-order, non-linear differential equation with an initial condition. Let's break down the process step-by-step:

1. Identifying the Type of Equation

The equation is a first-order differential equation because it involves the first derivative of the dependent variable y with respect to the independent variable t. It is also non-linear because of the terms involving powers of y greater than 1.

2. Rearranging the Equation

We can rearrange the equation to make it easier to solve:

(3y^2 - t^2y^5)dy/dt = -t^2y^4

Dividing both sides by (3y^2 - t^2y^5) and y^4 gives:

dy/(3y^2 - t^2y^5) = -t^2 dt / y^4

3. Integrating Both Sides

Now, we can integrate both sides of the equation:

∫ dy/(3y^2 - t^2y^5) = ∫ -t^2 dt / y^4

The left-hand side integral can be solved using partial fractions, and the right-hand side is a simple power rule integration.

4. Solving the Integrals

Left-hand Side:

Factor out y^2 from the denominator:
```
∫ dy/(y^2(3 - t^2y^3))
```

Apply partial fraction decomposition:

A/y^2 + B/(3 - t^2y^3) = 1/(y^2(3 - t^2y^3))

Solve for A and B:
```
A = 1/3,  B = t^2/3
```

Integrate:

∫ (1/3y^2 + t^2/(3(3 - t^2y^3))) dy = -1/(3y) - (1/9)ln|3 - t^2y^3| + C1

Right-hand Side:

∫ -t^2 dt / y^4 = t^3/(3y^4) + C2

5. Combining the Results

Now we have:

-1/(3y) - (1/9)ln|3 - t^2y^3| + C1 = t^3/(3y^4) + C2

6. Applying the Initial Condition

We are given the initial condition y(1) = 1. Substituting this into the equation, we can solve for the constant of integration (C1 - C2):

-1/3 - (1/9)ln|2| + C1 = 1/3 + C2

Solving for C1 - C2:

C1 - C2 = 2/3 + (1/9)ln|2|

7. Solving for the Explicit Solution

Substitute the value of C1 - C2 back into the equation and simplify to obtain the explicit solution for y in terms of t. This may involve complex algebraic manipulations and may not be possible to express y explicitly in terms of t.

Important Note: The explicit solution may be difficult to obtain and may involve implicit functions or numerical methods.

8. Verifying the Solution

Once you've found the explicit solution (or implicit form), you can verify it by plugging it back into the original differential equation and checking if it satisfies the equation and the initial condition.

This process provides a framework for solving the given differential equation with an initial condition. The exact solution will depend on the specific algebraic manipulations and techniques used to solve the integrals and simplify the result.