2x dy/dx+y=6x^3
How to solve ?
2xy' + y = 6x^3
y' + (1/2x)y = 3x^2
Look for an integrating factor:
u(x) = e^(∫1/(2x) dx) = √x
Now multiply by that and we have
√x y' + y/(2√x) = 3x^(5/2)
(√x y)' = 3x^(5/2)
√x y = 6/7 x^(7/2) + c
y = 6/7 x^3 + c/√x
Thanx a lot steve :)
To solve the differential equation 2x(dy/dx) + y = 6x^3, you can follow these steps:
Step 1: Rewrite the differential equation in its standard form.
The standard form for a first-order linear differential equation is: dy/dx + P(x)y = Q(x), where P(x) and Q(x) are functions of x. In this case, by dividing through by 2x, we obtain:
(dy/dx) + (1/2x)y = 3x^2
Step 2: Identify the integrating factor.
To find the integrating factor (IF), we need to multiply both sides of the equation by an appropriate function, such that the left-hand side becomes the derivative of the product of the integrating factor and y. In this case, the integrating factor is e^(∫(1/2x)dx), which simplifies to e^(1/2ln|x|) = e^(ln|x|^1/2) = e^(ln√|x|) = √|x|.
Step 3: Multiply the differential equation by the integrating factor.
By multiplying both sides of the equation by the integrating factor, we have:
√|x|(dy/dx) + (√|x|/2x)y = 3x^2√|x|
Simplifying this gives:
√|x| dy/dx + (1/2)y = 3x^2√|x|
Step 4: Apply the product rule to the left-hand side.
Differentiate the product of the integrating factor and y using the product rule:
d/dx (√|x|y) = (dy/dx)√|x| + y(1/2x)
Step 5: Substitute the derived expressions back into the differential equation.
Substituting the derived expressions from Steps 3 and 4 back into the original differential equation, we get:
(dy/dx)√|x| + y(1/2x) + (1/2)y = 3x^2√|x|
Step 6: Simplify the equation.
Consolidate the terms involving dy/dx and dy on the left-hand side:
(dy/dx)√|x| + (1/2x)y + (1/2)y = 3x^2√|x|
Step 7: Solve the equation.
Separate the variables by moving the terms involving dy/dx to one side and the terms involving y to the other side:
(dy/dx)√|x| = -[(1/2x)y + (1/2)y - 3x^2√|x|]
Step 8: Integrate both sides.
Integrating both sides with respect to x yields:
∫(dy/dx)√|x| dx = ∫-[(1/2x)y + (1/2)y - 3x^2√|x|] dx
Step 9: Simplify and integrate.
Solve the integrals on both sides of the equation.
∫(dy/dx)√|x| dx = ∫-[(1/2x)y + (1/2)y - 3x^2√|x|] dx
∫(√|x| dy) = -∫(1/2x)y dx - ∫(1/2)y dx + ∫3x^2√|x| dx
∫(√|x| dy) = -(1/2)∫(1/x)y dx - (1/2)∫y dx + 3∫x^2√|x| dx
Step 10: Evaluate the integrals.
Integrate each term on the left-hand side:
∫(√|x| dy) = y√|x| + constant
And integrate each term on the right-hand side:
-(1/2)∫(1/x)y dx = -(1/2)∫(y/x) dx = -(1/2)y ln|x| + constant
-(1/2)∫y dx = -(1/2)xy + constant
3∫x^2√|x| dx = 3(2/5)x^(5/2)√|x| + constant
Step 11: Combine all the terms.
Substitute the integrals back into the original equation:
y√|x| + constant = -(1/2)y ln|x| - (1/2)xy + 3(2/5)x^(5/2)√|x| + constant
Combine like terms:
y√|x| +(1/2)y ln|x| +(1/2)xy - 3(2/5)x^(5/2)√|x| = constant
This is the general solution to the given differential equation.
To solve the given differential equation:
Step 1: Identify the type of differential equation.
The given equation is a linear first-order ordinary differential equation. This is because it is of the form: dy/dx + P(x)y = Q(x), where P(x) = 1 and Q(x) = 6x^3.
Step 2: Find the integrating factor.
Multiply the entire equation by the integrating factor, which is given by the exponential of the integral of P(x)dx. In this case, the integrating factor is e^(∫P(x)dx) = e^(∫1 dx) = e^x.
Step 3: Rewrite the equation using the integrating factor.
Multiplying the given equation by the integrating factor e^x, you get:
e^x(2x dy/dx + y) = e^x(6x^3).
Step 4: Simplify the equation.
Distribute e^x on the left-hand side of the equation:
2x(e^x dy/dx) + ye^x = 6x^3e^x.
Step 5: Rewrite the left-hand side using the product rule.
Apply the product rule to the first term on the left-hand side:
(d/dx)(x^2e^x dy/dx) + ye^x = 6x^3e^x.
Step 6: Integrate both sides of the equation.
Integrate both sides of the equation with respect to x:
∫(d/dx)(x^2e^x dy/dx) dx + ∫ye^x dx = ∫6x^3e^x dx.
On the left-hand side, the first term can be simplified using the fundamental theorem of calculus:
x^2e^x dy/dx = ∫2x(e^x dy/dx) dx = ∫2x(6x^3e^x) dx.
On the right-hand side, ∫6x^3e^x dx can be easily evaluated using integration techniques.
Step 7: Solve for y.
Once you have integrated both sides of the equation, you will be left with an equation involving only y and x. Solve this equation using algebraic manipulations to isolate y.
This completes the solution process for the given differential equation.