Question
Solve the differential equation
y=6x2+12x+62x3+3x2+C,C∈R
Evaluate
(x+1)dxdy+2y=x
Multiply both sides
((x+1)dxdy+2y)×x+11=x×x+11
Apply the distributive property
(x+1)dxdy×x+11+2y×x+11=x×x+11
Multiply the terms
dxdy+2y×x+11=x×x+11
Multiply the terms
dxdy+x+12y=x×x+11
Multiply the terms
dxdy+x+12y=x+1x
Rewrite the expression
dxdy+x+12×y=x+1x
Since the equation is written in standard form, determine the functions P(x) and Q(x)
P(x)=x+12Q(x)=x+1x
Insert the function P(x)=x+12 into the formula for the integrating factor u(x)
u(x)=e∫x+12dxQ(x)=x+1x
Evaluate the integral
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![solution-arrow](../../wwwstatic/images/solution-arrow-down-1.svg)
Evaluate
∫x+12dx
Use the property of integral ∫kf(x)dx=k∫f(x)dx
2×∫x+11dx
Use ∫ax+b1dx=a1lnax+b to evaluate the integral
2ln(x+1)
u(x)=e2ln(x+1)Q(x)=x+1x
Rewrite the expression
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![solution-arrow](../../wwwstatic/images/solution-arrow-down-1.svg)
Evaluate
e2ln(x+1)
Use nlnu=lnun to transform the expression
eln((x+1)2)
Use elnu=u to transform the expression
(x+1)2
u(x)=(x+1)2Q(x)=x+1x
Insert the integrating factor u(x) and the function Q(x) into the general solution formula
y=(x+1)21×∫x+1x×(x+1)2dx
Calculate
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![solution-arrow](../../wwwstatic/images/solution-arrow-down-1.svg)
Evaluate
∫x+1x×(x+1)2dx
Multiply the terms
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![solution-arrow](../../wwwstatic/images/solution-arrow-down-2.svg)
Multiply the terms
x+1x×(x+1)2
Cancel out the common factor x+1
x(x+1)
Apply the distributive property
x×x+x×1
Multiply the terms
x2+x×1
Any expression multiplied by 1 remains the same
x2+x
∫x2+xdx
Use the property of integral ∫f(x)±g(x)dx=∫f(x)dx±∫g(x)dx
∫x2dx+∫xdx
Evaluate the integral
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![solution-arrow](../../wwwstatic/images/solution-arrow-down-2.svg)
Evaluate
∫x2dx
Use ∫xndx=n+1xn+1,n=−1 to evaluate the integral
3x3
Simplify
31x3
31x3+∫xdx
Evaluate the integral
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![solution-arrow](../../wwwstatic/images/solution-arrow-down-2.svg)
Evaluate
∫xdx
Use ∫xndx=n+1xn+1,n=−1 to evaluate the integral
2x2
Simplify
21x2
31x3+21x2
Add the constant of integral C
31x3+21x2+C,C∈R
y=(x+1)21×(31x3+21x2+C),C∈R
Calculate
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![solution-arrow](../../wwwstatic/images/solution-arrow-down-1.svg)
Evaluate
(x+1)21×(31x3+21x2+C)
Rewrite the expression
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![solution-arrow](../../wwwstatic/images/solution-arrow-down-2.svg)
Evaluate
31x3+21x2+C
Rewrite the expression
62x3+3x2+C
Reduce fractions to a common denominator
62x3+3x2+6C×6
Calculate
62x3+3x2+66C
Add or subtract the terms
62x3+3x2+6C
Since C is a constant,replace the 6C with the constant C
62x3+3x2+C
(x+1)21×62x3+3x2+C
Multiply the terms
(x+1)2×62x3+3x2+C
Use the commutative property to reorder the terms
6(x+1)22x3+3x2+C
y=6(x+1)22x3+3x2+C,C∈R
Solution
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![solution-arrow](../../wwwstatic/images/solution-arrow-down-1.svg)
Evaluate
y=6(x+1)22x3+3x2+C
Calculate
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![solution-arrow](../../wwwstatic/images/solution-arrow-down-2.svg)
Evaluate
6(x+1)2
Calculate
6(x2+2x+1)
Calculate
6x2+12x+6
y=6x2+12x+62x3+3x2+C
y=6x2+12x+62x3+3x2+C,C∈R
Show Solution
![solution-arrow](data:image/svg+xml;base64,PHN2ZyB3aWR0aD0iMTciIGhlaWdodD0iMTYiIHZpZXdCb3g9IjAgMCAxNyAxNiIgZmlsbD0ibm9uZSIgeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzIwMDAvc3ZnIj4KPHBhdGggZD0iTTguNSAyLjY2Nzk3TDguNSAxMi42NjgiIHN0cm9rZT0id2hpdGUiIHN0cm9rZS13aWR0aD0iMS41IiBzdHJva2UtbGluZWNhcD0icm91bmQiIHN0cm9rZS1saW5lam9pbj0icm91bmQiLz4KPHBhdGggZD0iTTEyLjUgOC42Njc5N0w4LjUgMTIuNjY4TDQuNSA4LjY2Nzk3IiBzdHJva2U9IndoaXRlIiBzdHJva2Utd2lkdGg9IjEuNSIgc3Ryb2tlLWxpbmVjYXA9InJvdW5kIiBzdHJva2UtbGluZWpvaW49InJvdW5kIi8+Cjwvc3ZnPgo=)