Solve the following differential equation:dy/dx = 3xy + x^3e^(x^2)?

Answer 1

To solve the differential equation dy/dx = 3xy + x^3e^(x^2), we can use the method of integrating factors. Here's the step-by-step solution:

Step 1: Recognize the form of the differential equation as a first-order linear differential equation of the form dy/dx + P(x)y = Q(x), where P(x) = 3x and Q(x) = x^3e^(x^2).

Step 2: Multiply the entire equation by the integrating factor, which is the exponential of the integral of P(x). In this case, the integrating factor is e^(∫3x dx) = e^(3x^2/2).

Multiplying the equation by the integrating factor gives us:

e^(3x^2/2) dy/dx + 3x e^(3x^2/2) y = x^3 e^(3x^2/2) * e^(x^2).

Step 3: Notice that the left side of the equation can be simplified using the product rule for differentiation. The derivative of (e^(3x^2/2) y) with respect to x is given by d/dx (e^(3x^2/2) y) = e^(3x^2/2) dy/dx + 3x e^(3x^2/2) * y.

Using this, the equation becomes:

d/dx (e^(3x^2/2) y) = x^3 e^(4x^2/2).

Step 4: Integrate both sides of the equation with respect to x:

∫d/dx (e^(3x^2/2) y) dx = ∫x^3 e^(4x^2/2) dx.

This simplifies to:

e^(3x^2/2) y = ∫x^3 e^(2x^2) dx.

Step 5: Evaluate the integral on the right side of the equation:

To solve the integral, we can use integration by parts. Let u = x^2 and dv = xe^(2x^2) dx. Then, du = 2x dx and v = (1/4) * e^(2x^2).

Using integration by parts, the integral becomes:

∫x^3 e^(2x^2) dx = (1/4) x^2 e^(2x^2) - (1/4) ∫2x^2 * e^(2x^2) dx.

Notice that we have another integral on the right side that is similar to the original integral. We can repeat the integration by parts process until we have an integral that we can solve.

After integrating by parts twice, the integral becomes:

∫x^3 e^(2x^2) dx = (1/4) x^2 e^(2x^2) - (1/8) x e^(2x^2) + (1/16) ∫e^(2x^2) dx.

The remaining integral, ∫e^(2x^2) dx, is a standard Gaussian integral and cannot be expressed in elementary functions. Therefore, we can denote it as √(π/8) * erf(x√2).

Step 6: Substitute the evaluated integral back into the equation:

e^(3x^2/2) y = (1/4) x^2 e^(2x^2) - (1/8) x * e^(2

x^2) + (1/16) √(π/8) erf(x√2).

Step 7: Solve for y by dividing both sides of the equation by e^(3x^2/2):

y = (1/4) x^2 e^(x^2) - (1/8) x e^(x^2) + (1/16) √(π/8) e^(-3x^2/2) * erf(x√2).

Thus, the solution to the given differential equation is y = (1/4) x^2 e^(x^2) - (1/8) x e^(x^2) + (1/16) √(π/8) e^(-3x^2/2) * erf(x√2).

Related More Question

Visit : maths assignment help