Calculus Examples

Solve the Differential Equation
dydx=yx+yxydydx=yx+yxy
Step 1
Rewrite the differential equation as a function of yxyx.
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Step 1.1
Assume y2=yy2=y.
dydx=yx+y2xydydx=yx+y2xy
Step 1.2
Combine y2y2 and xyxy into a single radical.
dydx=yx+y2xydydx=yx+y2xy
Step 1.3
Reduce the expression y2xyy2xy by cancelling the common factors.
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Step 1.3.1
Factor yy out of y2y2.
dydx=yx+yyxydydx=yx+yyxy
Step 1.3.2
Factor yy out of xyxy.
dydx=yx+yyyxdydx=yx+yyyx
Step 1.3.3
Cancel the common factor.
dydx=yx+yyyx
Step 1.3.4
Rewrite the expression.
dydx=yx+yx
dydx=yx+yx
dydx=yx+yx
Step 2
Let V=yx. Substitute V for yx.
dydx=V+V
Step 3
Solve V=yx for y.
y=Vx
Step 4
Use the product rule to find the derivative of y=Vx with respect to x.
dydx=xdVdx+V
Step 5
Substitute xdVdx+V for dydx.
xdVdx+V=V+V
Step 6
Solve the substituted differential equation.
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Step 6.1
Separate the variables.
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Step 6.1.1
Solve for dVdx.
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Step 6.1.1.1
Move all terms not containing dVdx to the right side of the equation.
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Step 6.1.1.1.1
Subtract V from both sides of the equation.
xdVdx=V+V-V
Step 6.1.1.1.2
Combine the opposite terms in V+V-V.
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Step 6.1.1.1.2.1
Subtract V from V.
xdVdx=0+V
Step 6.1.1.1.2.2
Add 0 and V.
xdVdx=V
xdVdx=V
xdVdx=V
Step 6.1.1.2
Divide each term in xdVdx=V by x and simplify.
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Step 6.1.1.2.1
Divide each term in xdVdx=V by x.
xdVdxx=Vx
Step 6.1.1.2.2
Simplify the left side.
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Step 6.1.1.2.2.1
Cancel the common factor of x.
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Step 6.1.1.2.2.1.1
Cancel the common factor.
xdVdxx=Vx
Step 6.1.1.2.2.1.2
Divide dVdx by 1.
dVdx=Vx
dVdx=Vx
dVdx=Vx
dVdx=Vx
dVdx=Vx
Step 6.1.2
Multiply both sides by 1V.
1VdVdx=1VVx
Step 6.1.3
Cancel the common factor of V.
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Step 6.1.3.1
Cancel the common factor.
1VdVdx=1VVx
Step 6.1.3.2
Rewrite the expression.
1VdVdx=1x
1VdVdx=1x
Step 6.1.4
Rewrite the equation.
1VdV=1xdx
1VdV=1xdx
Step 6.2
Integrate both sides.
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Step 6.2.1
Set up an integral on each side.
1VdV=1xdx
Step 6.2.2
Integrate the left side.
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Step 6.2.2.1
Apply basic rules of exponents.
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Step 6.2.2.1.1
Use nax=axn to rewrite V as V12.
1V12dV=1xdx
Step 6.2.2.1.2
Move V12 out of the denominator by raising it to the -1 power.
(V12)-1dV=1xdx
Step 6.2.2.1.3
Multiply the exponents in (V12)-1.
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Step 6.2.2.1.3.1
Apply the power rule and multiply exponents, (am)n=amn.
V12-1dV=1xdx
Step 6.2.2.1.3.2
Combine 12 and -1.
V-12dV=1xdx
Step 6.2.2.1.3.3
Move the negative in front of the fraction.
V-12dV=1xdx
V-12dV=1xdx
V-12dV=1xdx
Step 6.2.2.2
By the Power Rule, the integral of V-12 with respect to V is 2V12.
2V12+C1=1xdx
2V12+C1=1xdx
Step 6.2.3
The integral of 1x with respect to x is ln(|x|).
2V12+C1=ln(|x|)+C2
Step 6.2.4
Group the constant of integration on the right side as C.
2V12=ln(|x|)+C
2V12=ln(|x|)+C
Step 6.3
Solve for V.
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Step 6.3.1
Divide each term in 2V12=ln(|x|)+C by 2 and simplify.
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Step 6.3.1.1
Divide each term in 2V12=ln(|x|)+C by 2.
2V122=ln(|x|)2+C2
Step 6.3.1.2
Simplify the left side.
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Step 6.3.1.2.1
Cancel the common factor.
2V122=ln(|x|)2+C2
Step 6.3.1.2.2
Divide V12 by 1.
V12=ln(|x|)2+C2
V12=ln(|x|)2+C2
Step 6.3.1.3
Simplify the right side.
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Step 6.3.1.3.1
Simplify each term.
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Step 6.3.1.3.1.1
Rewrite ln(|x|)2 as 12ln(|x|).
V12=12ln(|x|)+C2
Step 6.3.1.3.1.2
Simplify 12ln(|x|) by moving 12 inside the logarithm.
V12=ln(|x|12)+C2
V12=ln(|x|12)+C2
V12=ln(|x|12)+C2
V12=ln(|x|12)+C2
Step 6.3.2
Raise each side of the equation to the power of 2 to eliminate the fractional exponent on the left side.
(V12)2=(ln(|x|12)+C2)2
Step 6.3.3
Simplify the left side.
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Step 6.3.3.1
Simplify (V12)2.
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Step 6.3.3.1.1
Multiply the exponents in (V12)2.
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Step 6.3.3.1.1.1
Apply the power rule and multiply exponents, (am)n=amn.
V122=(ln(|x|12)+C2)2
Step 6.3.3.1.1.2
Cancel the common factor of 2.
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Step 6.3.3.1.1.2.1
Cancel the common factor.
V122=(ln(|x|12)+C2)2
Step 6.3.3.1.1.2.2
Rewrite the expression.
V1=(ln(|x|12)+C2)2
V1=(ln(|x|12)+C2)2
V1=(ln(|x|12)+C2)2
Step 6.3.3.1.2
Simplify.
V=(ln(|x|12)+C2)2
V=(ln(|x|12)+C2)2
V=(ln(|x|12)+C2)2
V=(ln(|x|12)+C2)2
Step 6.4
Simplify the constant of integration.
V=(ln(|x|12)+C)2
V=(ln(|x|12)+C)2
Step 7
Substitute yx for V.
yx=(ln(|x|12)+C)2
Step 8
Solve yx=(ln(|x|12)+C)2 for y.
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Step 8.1
Multiply both sides by x.
yxx=(ln(|x|12)+C)2x
Step 8.2
Simplify.
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Step 8.2.1
Simplify the left side.
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Step 8.2.1.1
Cancel the common factor of x.
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Step 8.2.1.1.1
Cancel the common factor.
yxx=(ln(|x|12)+C)2x
Step 8.2.1.1.2
Rewrite the expression.
y=(ln(|x|12)+C)2x
y=(ln(|x|12)+C)2x
y=(ln(|x|12)+C)2x
Step 8.2.2
Simplify the right side.
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Step 8.2.2.1
Reorder factors in (ln(|x|12)+C)2x.
y=x(ln(|x|12)+C)2
y=x(ln(|x|12)+C)2
y=x(ln(|x|12)+C)2
y=x(ln(|x|12)+C)2
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 [x2  12  π  xdx ] 
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