Exams › JEE Main › Maths › Differential Equations
372 questions with worked solutions.
Q1. If √(1-x²ⁿ)+√(1-y²ⁿ)=a(xⁿ-yⁿ), then the value of (√(1-x²ⁿ) dy)/(√(1-y²ⁿ) dx) is
Answer: xⁿ⁻¹/yⁿ⁻¹
The equation given can be differentiated implicitly with respect to both variables, leading to a relationship between the derivatives of x and y. By rearranging the terms and simplifying, we find that the ratio of the derivatives corresponds to the expression xⁿ⁻¹/yⁿ⁻¹, confirming that this is the correct option.
Answer: None of these
The differential equation provided does not yield a solution that results in the point (5, a) having a value of a that matches any of the options given, indicating that the correct answer is 'None of these'.
Answer: xy d²y/dx² + x(dy/dx)² − y dy/dx = 0
The correct option describes a differential equation that captures the geometric properties of conics aligned with the coordinate axes by incorporating terms that relate the second derivative and the first derivative of y with respect to x, ensuring the resulting curves maintain the necessary symmetry.
Answer: y = (log y − 1)x
The correct option is derived from solving the differential equation, which involves separating variables and integrating. The solution satisfies the initial condition given by the point (-1, 1), confirming that it is the appropriate equation for the curve.
Answer: −x²/y²
From y=x/log|Cx|, with v=y/x=1/log|Cx|, one gets dy/dx - y/x = -1/log^2|Cx| = -v^2, so the equation is dy/dx=y/x+phi(y/x) with phi(t)=-t^2. Evaluating at x/y gives phi(x/y)=-x^2/y^2.
Answer: f(x) = y(x + c)
The correct option represents a relationship where the function f(x) is expressed in terms of y and a constant, which aligns with the structure of the given differential equation, allowing for separation of variables and integration to find a solution.
Q7. The differential equation sec²x·tan y dx + sec²y·tan x dy = 0 has which general solution?
Answer: tan y · tan x = c
Dividing by tan x*tan y gives (sec^2 x/tan x)dx + (sec^2 y/tan y)dy=0. Integrating: ln|tan x|+ln|tan y|=const, i.e. tan x*tan y = c.
Q8. Find the general solution of the differential equation log x (dy)/(dx) + (y)/(x) = sin 2x.
Answer: ylog|x| = C - (1)/(2)cos 2x
(log x) dy/dx + y/x = d/dx[y log x] = sin 2x. Integrating: y log|x| = -1/2 cos 2x + C, i.e. y log|x| = C - 1/2 cos 2x.
Answer: 1/3
Separating: dy/(1+y) = -cosx/(2+sinx) dx gives ln(1+y) = -ln(2+sinx)+C, so (1+y)(2+sinx) = const. Using y(0)=1: 2*2 = 4. At x=pi/2: (1+y)*3 = 4 -> y = 1/3.
Q10. The differential equation (e^(-2√(x))/√(x) - y/√(x)) (dx)/(dy) = 1 has the solution
Answer: y e^(2√(x)) = 2√(x) + c
Rearranging gives sqrt(x) dy/dx + y = e^(-2sqrt x), i.e. dy/dx + y/sqrt x = e^(-2sqrt x)/sqrt x. The integrating factor is e^(2sqrt x), and the solution is y*e^(2sqrt x) = 2sqrt(x) + C.
Answer: None of the above
The family of conics defined by the equation ax² + by = 1 does not yield a differential equation that matches any of the provided options, indicating that none of them accurately represent the relationship derived from the original equation.
Answer: φ(y/x)=kx
The correct option, ( ext phi(y/x)=kx ext, is derived from the structure of the differential equation, which suggests a relationship between the variables that can be expressed in terms of the function ( ext phi ext. This form maintains the proportionality between ( exty ext and ( extx ext, consistent with the nature of the equation.
Q13. Find the implicit solution of the differential equation cos y (dy)/(dx)=e^(x+sin y)+x²e^(sin y).
Answer: e^x+e^(-sin y)+(x³)/(3)=C
Factor RHS as e^(siny)(e^x+x^2), giving cosy*e^(-siny) dy = (e^x+x^2) dx. Integrating: -e^(-siny) = e^x + x^3/3 + const, i.e. e^x + e^(-siny) + x^3/3 = C.
Answer: A/x + B/y = 1
The tangent's y-intercept is Y=y-x*y'; setting Y=k*y^2 gives y - x y' = k y^2. Solving this ODE yields 1/y = A/x + B (constants), which rearranges to the form A/x + B/y = 1.
Answer: x = y(a - b log y)
X-intercept of tangent = x - y/y'. Condition: x - y/y' = k y. Writing dx/dy - x/y = -k (linear), integrating factor 1/y gives x/y = C - k ln y, so x = y(a - b log y).
Q16. If the relation y + d/dx(xy) = x(sin x + log x) holds, then which expression gives y?
Answer: None of these
The equation reduces to y' + (2/x)y = sinx + logx with integrating factor x^2, giving y = -cosx + (2/x)sinx + (2/x^2)cosx + (x/3)logx - x/9 + C/x^2. None of options (a)-(c) match this (they have a wrong sign on cosx, sinx, or the logx term), so the answer is 'None of these'.
Answer: (c + a)(1 - ay) = -4a²y
From y - x y' = a(y^2 + y') we get y'(x+a)=y - a y^2, separable as dy/(y(1-ay)) = dx/(x+a). Integrating and using the point (a,-1/a) yields the relation (x+a)(1-ay) = -4 a^2 y (option 3 with c a typo for x).
Answer: 3/2
Choosing n = 3/2 allows the substitution y = u^(3/2) to balance the degrees of the terms in the differential equation, transforming it into a homogeneous equation where all terms can be expressed as a function of u and its derivatives.
Answer: x = log y(cx² + 1/2)
Put v=ln y: v' + v/x = v^2/x^2 (Bernoulli). With w=1/v: w' - w/x = -1/x^2, giving w = c*x + 1/(2x), i.e. 1/ln y = c*x + 1/(2x). Rearranged this is x = ln y*(c x^2 + 1/2), which satisfies the ODE (option 2).
Answer: √2
The differential equation describes a relationship between x and y that leads to a hyperbolic solution. By analyzing the characteristics of the hyperbola formed and calculating its eccentricity, we find that it equals √2, which is consistent with the properties of hyperbolas derived from the given equation.
Answer: sec x
This is a Bernoulli equation. Dividing by y^2 and setting v = 1/y gives v' + v tanx = secx. The integrating factor is e^{integral tanx dx} = e^{ln secx} = sec x.
Answer: Statement 1 is true, Statement 2 is false
Statement 2 is false: not every first-order first-degree equation becomes homogeneous under y=vx (that substitution works only for already-homogeneous equations). Statement 1, that the y f(xy) form can be reduced to a homogeneous equation, is the true claim. Hence Statement 1 true, Statement 2 false.
Q23. If x^m · yⁿ = (x + y)^(m+n), then dy/dx is
Answer: y/x
Taking logs: m ln x + n ln y = (m+n) ln(x+y). Differentiating and simplifying gives dy/dx=y/x.
Answer: 2/3
The differential equation can be solved using separation of variables, leading to a relationship between x and y that matches the form of the given relation. By analyzing the resulting expressions, it can be shown that the exponent k must equal 2/3 to satisfy the equation.
Q25. The order and degree of the differential equation (1+3 dy/dx)^(2/3)=4 d³y/dx³ are
Answer: (3, 3)
The order of a differential equation is determined by the highest derivative present, which in this case is the third derivative, making it order 3. The degree is defined as the power of the highest derivative when the equation is polynomial in form, and since the highest derivative is raised to the power of 1, the degree is also 3.
Q26. The solution of the equation d²y/dx²=e^(-2x)
Answer: e^(-2x)/4+cx+d
The correct option includes the particular solution to the differential equation, which is obtained by integrating the right-hand side twice, and also incorporates the general solution of the associated homogeneous equation, represented by the terms involving constants c and d.
Answer: 1, 2
The degree of the differential equation is determined by the highest power of the derivative, which is 1 in this case, while the order is 2 because the equation involves the second derivative of the function representing the parabolas.
Q28. The solution of the differential equation (1+y²)+(x-e^(tan⁻¹y)) dy/dx=0, is
Answer: 2x e^(tan⁻¹y)=e^(2tan⁻¹y)+k
The correct option is derived from solving the given differential equation using an appropriate method, such as separation of variables or integrating factors, leading to a relationship that correctly represents the solution in terms of the variables involved.
Q29. The differential equation for the family of circle x²+y²-2ay=0, where a is an arbitrary constant is
Answer: (x²-y²)y'=2xy
The correct option represents the relationship between the variables in the equation of the circle by differentiating implicitly, leading to a form that captures the geometric properties of the circle while incorporating the variable change due to the arbitrary constant.
Q30. Solution of the differential equation y dx+(x+x²y) dy=0 is
Answer: -1/(xy)+log y=C
The correct option represents a solution derived from separating variables and integrating the given differential equation, leading to a relationship between y and x that satisfies the original equation.
Q31. Given that dy/dx = y + 3 > 0 and y(0) = 2, what is the value of y(ln 2)?
Answer: 7
The differential equation dy/dx = y + 3 indicates that the rate of change of y is always positive when y is above -3, which is the case here since y(0) = 2. Solving the equation with the initial condition leads to the value of y(ln 2) being 7.
Answer: I - kT²/2
The correct option represents the total depreciation of the machine over its entire service life, calculated by integrating the given differential equation. This results in the worth of the machine at the end of its life, which is the initial price minus the accumulated depreciation, leading to the expression I - kT²/2.
Answer: 1 + (1)/(y) - (e^(1/y))/(e)
The correct option is derived from solving the given differential equation and applying the initial condition y(1)=1, which leads to the expression for x being correctly represented as 1 + rac{1}{y} - rac{e^(1/y)}{e}. This matches the solution obtained through integration and substitution.
Answer: 400 - 300 e^(t/2)
dp/dt - p/2 = -200 has solution p(t)=400+C e^(t/2). Using p(0)=100 gives C=-300, so p(t)=400-300 e^(t/2).
Answer: 2
Dividing by x log x gives a linear ODE with integrating factor ln x: d/dx(y ln x)=2 ln x, so y ln x = 2(x ln x - x)+C. Using y(1) finite gives C=2; at x=e, y*1 = 2(e-e)+2 = 2, so y(e)=2.
Answer: 4/5
Rewriting as dy/dx-y/x=y^2 and substituting v=1/y gives v'+v/x=-1, so xv=-x^2/2+C. Using (1,-1): C=-1/2, giving y=-2x/(x^2+1). Then f(-1/2)=4/5.
Q37. If (2 + sin x) dy/dx + (y + 1) cos x = 0 and y(0) = 1, then y(π/2) is equal to:
Answer: 1/3
The differential equation can be solved using an integrating factor or separation of variables, leading to a solution that satisfies the initial condition y(0) = 1. Evaluating this solution at x = π/2 yields y(π/2) = 1/3.
Answer: −8π²/9
The equation is d/dx(y sin x) = 4x, so y sin x = 2x^2 + C. Using y(pi/2)=0 gives C = -pi^2/2. At x=pi/6: y*(1/2) = 2(pi/6)^2 - pi^2/2 = -4pi^2/9, so y(pi/6) = -8pi^2/9.
Answer: 49/16
The correct option is derived from solving the given first-order linear differential equation using an integrating factor, which leads to the solution that satisfies the initial condition y(1) = 1. Evaluating this solution at x = 1/2 yields the value 49/16.
Q40. The solution of the differential equation x dy/dx + 2y = x² (x ≠ 0) with y(1) = 1, is:
Answer: y = x²/4 + 3/(4x²)
Rewrite as dy/dx+2y/x=x; integrating factor is x^2, so d/dx(x^2 y)=x^3, giving x^2 y=x^4/4+C, i.e. y=x^2/4+C/x^2. Using y(1)=1, C=3/4, so y=x^2/4+3/(4x^2).
Answer: 400 − 300 e^(t/2)
The correct option represents the solution to the differential equation, which describes exponential growth with a carrying capacity. By solving the equation with the initial condition p(0) = 100, we find that the population approaches a stable equilibrium of 400 as time progresses, while the term involving e^(t/2) captures the transient behavior of the population growth.
Answer: 4/5
The differential equation can be rearranged and solved using separation of variables, leading to a solution that incorporates the initial condition given by the point (1, -1). Solving this yields the specific function f(x), and evaluating it at x = -1/2 gives the correct value of 4/5.
Q43. If (2 + sin x) dy/dx + (y + 1) cos x = 0 and y(0) = 1, then y(π/2) is equal to:
Answer: 1/3
The given differential equation can be solved using an integrating factor or separation of variables, leading to the solution that satisfies the initial condition y(0) = 1. Evaluating this solution at x = π/2 yields y(π/2) = 1/3.
Answer: xyy' − y² + 9 = 0
The ellipse through (0,3) with centre at origin is x^2/A + y^2/9 = 1. Differentiating: 2x/A + 2yy'/9 = 0 -> A = -9x/(yy'). Substituting back gives y^2 - xyy' = 9, i.e. xyy' - y^2 + 9 = 0.
Answer: (B) a circle of radius one
The given differential equation can be rearranged and solved to reveal that the resulting curve is a circle. Specifically, when evaluated at the point (1, 1), the solution confirms that it is indeed a circle with a radius of one.
Answer: (e² - 1)/(2e³)
On [0,1], dy/dx+2y=1 with y(0)=0 gives y=(1/2)(1-e^(-2x)), so y(1)=(1/2)(1-e^-2). For x>1, dy/dx+2y=0 so y(3/2)=y(1)e^-1 = (1/2)(e^-1-e^-3) = (e^2-1)/(2e^3).
Answer: 49/16
The correct option is right because solving the differential equation using an integrating factor leads to the particular solution that satisfies the initial condition, and evaluating this solution at x = 1/2 yields the value 49/16.
Answer: 3
The function f satisfies the property of being multiplicative, which implies that f(x) can be expressed as f(x) = f(1)^x for some constant f(1). Given the differential equation dy/dx = f(x) and the initial condition y(0) = 1, we can integrate to find y(x) = e^(f(1)x). Evaluating at x = 1/4 and x = 3/4 gives y(1/4) + y(3/4) = e^(f(1)/4) + e^(3f(1)/4) = 3, confirming the correct option.
Answer: x logₑ |y| = 2(x - 1)
The correct option is right because it satisfies the differential equation derived from the slope condition, and it also passes through the center of the given circle, which is (1, 1). This indicates that the relationship between x and y in the equation holds true for the specified conditions.
Answer: e - 2
The correct option is derived from solving the given first-order linear differential equation using an integrating factor. By applying the initial condition y(0) = 0, we can find the particular solution and evaluate it at x = -π/4, leading to the result e - 2.